Newsgroups: sprouts-theory From: Dan Hoey Date: Sun, 11 Jan 2009 10:25:47 -0800 (PST) Subject: Re: Sprouts notation Josh Purinton wrote: > I'm also a huge fan of the parenthesis idea. It's so obvious in > retrospect, but no one (besides Dan) thought of it. However, I'd > prefer to use angle brackets ("<" and ">") instead of parenthesis, > since I want to reserve parenthesis for more general purposes, such > as enclosing the whole game (see below). I suppose this makes sense, since parentheses are traditional for general grouping. The problems with HTML would arise in any case, since the alternative is to use angle brackets for repetitions. [ On my suggestion of separating boundaries with ".", regions with "/", and lands with "%"] > I agree, except I'd prefer to see "+" as the component ("land") > separator,since it's use for this purpose has a long history in > combinatorial game theory. I would retain "%" as the separator between > partial positions and the final position in which they are instantiated. That's fine, though I still want to be able to instantiate partial positions in a multiple-land position. It might even be worthwhile to allow a partial position to include multiple lands, so that the partial position ^x:x1 could be written ^x:x2+1 or even ^x:x2+*1. > > For nonplanar regions, I suggest a region should be suffixed with > > +n for T^n or -n for P^n. Since the region will always be > > followed by "/", "%", "!", or end-of-string, there is no > > restriction on multiple digits for n. > I'd like to avoid the use of plus and minus here, since they have > other connotations in sprouts notation. Perhaps instead we could write > Tn or Cn as a suffix. That sounds good. > For example, we would write (0.0+1<1>)T3. This is confusing, since the topology notation refers to a particular region. Each region has its own topology signifier (or none, if the region is planar). So is this supposed to mean 0.0T3+1<1>T3? I would rather not apply a topology signifier to all the regions in a position, since different regions usually end up with different topologies. > > Finally, I would like a construct for specifying the join of two > > partial positions. I propose using "=xyz=tuv" , where the first > > "=" replaces the preceding "%" character. So > > ^xyzw:xyzw1=xyzw=xywz refers to the (nonplanar) position > > ABCD1/ABDC1 > I don't understand this. Do you have a planar example? Also, what > about joining three partial positions? This notation could be used for joining an arbitrary number of partial positions. The reason for the equal sign notation is that a partial position is normally instantiated by mentioning the exterior site of a pivot that appears in the partial position. But if the exterior site of one partial position is internal to the other, this doesn't work. Perhaps the simplest case is the empty loop partial position ^xy^st^uv:xy!. Three of them could be put together in a cycle as ^xy^st^uv:xy=xsu=tvy!. This is just a long way of saying AB/BC/CA!. On further consideration, we may want to use partial positions only in cases where the external points appear in a single region. In this case, we canonicalize a position P by taking its region graph, R(P). The vertices of R(P) are the regions of P, and the edges of R(P) connect regions that share a pivot in P. We separate R(P) into its singly-connected components (SCCs): two vertices are in the same SCC if there are two disjoint paths between them. The SCCs form a tree. A tree is canonicalized by height: The height of a leaf node is zero, and a non-leaf node of degree N has height one greater than the maximum height of its N-1 neighbors of least height. A neighbor of equal or greater height is called the parent. When all heights are assigned, there will either be a root node of greatest height with no parent, or two nodes tied for greatest height, connected by an edge, which are each other's parents. We create a partial position for each SCC except the root (if any). The partial position's parameters are the pivots shared with the parent, and the partial position refers to the partial positions of which it is the parent. Of course, if two partial positions are isomorphic, only one need be defined. If there is a root, it is the position at the end of the partial positions. If there are two co-parents, then the equals-sign notation is used to express their connection. As an example, consider the position 0.A/AB/BC.DE/0.C/DF/EF!. The SCCs at height 0 are 0.A, 0.C, and DF/EF. The other two regions are co-parents. So the partial position notation would be ^x:0.x%^yz:yA/zA%^s:sx%^t:tx.yz=s=t!. --- Newsgroups: sprouts-theory From: Dan Hoey Date: Mon, 12 Jan 2009 16:46:20 -0500 Subject: Re: Sprouts notation I wrote: > As an example, consider the position 0.A/AB/BC.DE/0.C/DF/EF!. The > SCCs at height 0 are 0.A, 0.C, and DF/EF. The other two regions are > co-parents. So the partial position notation would be > ^x:0.x%^yz:yA/zA%^s:sx%^t:tx.yz=s=t!. On further consideration, the percent signs are superfluous. A partial position can be followed by a "^", introducing another partial position, an "=", introducing a coparent correspondence, or a ":" (instead of a "%"), introducing the root SCC. Then this position would be ^r:0.r^st:sA/tA^u:ru^v:rv.st=u=v!. If we are representing a multi-land position, I suggest that coparent pairs appear first (each as an "=...=..." clause) followed by the root SCCs, where all but the first root SCC can be introduced with a "+" instead of a ":". So 0.A/AB/0.B+0.A/AB/BC/0.C+0.A/AB/BC/CD/0.D+0.A/AB/BC/CD/DE/0.E would be written ^r:0.r^s:rs^t:st=s=s=t=t+rr+ss!. Of course, this would probably be placed in the database as r:2r^s:rs^t:st=s=s=t=t+rr+ss!, since ^r:0.r = ^r:2r . Note that coparent correspondences do not have topological signifiers (since they don't represent any new regions) but the regions of the SCCs may be suffixed with Tn or Cn for nonplanar topologies. This decomposition into SCCs is a first step toward graph isomorphism canonicalization. Dan --- Newsgroups: sprouts-theory From: Dan Hoey Date: Tue, 13 Jan 2009 19:38:42 -0500 Subject: Re: Sprouts notation Josh Purinton wrote: > If I understand the notation for joining multiple partial positions, > the syntax is > [partial position specifiers for sites abc...xyz...]=abc...=xyz... > and the meaning is that all the partial positions involving the sites > abc...xyz... are instantiated such that a & x are two sites of the > same spot, b & y are two sites of the same spot, c & z are two sites > of the same spot, etc. Clever. > See the attached diagram of your example ^xy^st^uv:xy=xsu=tvy. Yes, that's what the "=" notation means. I see I totally omitted explaining how I intended it to set up a matching between parameters, so thanks for your patient reconstruction. But if we use partial positions for simply-connected components, then that wouldn't be used because AB/BC/CA! is not simply-connected. In an SCC tree, the "=" notation is only used for co-roots. Those would be partial positions representing SCCs at the same height that appear on opposite sides of a boundary, like ^xy:xyAB/BC/CA=xy=xy! for ABCD/ABEF/CG/GD/EH/HF!. Dan --- Newsgroups: sprouts-theory From: Dan Hoey Date: Sat, 17 Jan 2009 13:51:19 -0500 Subject: Re: 17- lapinot@tuxfamily.org wrote: > Using Glop's new abilities for misere computation, we just managed to > compute that 17- is a win for the first player. Congratulations! Dan --- Newsgroups: sprouts-theory From: Dan Hoey Date: Sat, 17 Jan 2009 17:00:14 -0500 Subject: Re: 17- lapinot@tuxfamily.org wrote: > Using Glop's new abilities for misere computation, we just managed > to compute that 17- is a win for the first player. I hope the first player has a strategy that is isomorphic to an 11- strategy for the first few moves. I don't know how far you are from showing that the first player wins in misère Sprouts when the remainder (mod 6) is zero, four, or five, except that the first player wins the one-spot game and loses the four-spot game. Dan --- Making Light: Open thread 118 ::: January 18, 2009, 07:11 AM Xopher, thank you for helping us bring out our feelings for you. Whatever you think you were being a dork about, that's not you. You have my sympathy and love. If you have something to share about your father or your process, I'd like to hear it. When you are not preoccupied, you know what hogging the thread is --and isn't--as well as any of us. Thank you for making the thread. --- Making Light: Open thread 118 ::: January 18, 2009, 08:03 AM Greetings, friends, from the other side of the slough. At least it looks like I'm on my way out of it. The past three months have been very gray and dim, probably from a couple of friends' funerals, a couple of hospitalizations, the roosting of procrastinated stress, disrupted meds, and a cancer diagnosis. Things started brightening up about a week before the prostatectomy that I had scheduled in August, on the way down. It was about the latest I could get started and actually make it happen. As it turned out, things went really smoothly. Now twelve days post-op, I have every reason to expect a complete recovery. I hope it's not just gratuitous name-dropping to mention that I got knifed by Dr. Patrick C. Walsh, the man who revolutionized prostatectomy technique in the 1980s. Before him, the prognosis was 100% impotence, nearly 100% incontinence, and 0% effectiveness against cancer (the last due to late cancer detection and postponement of prostatectomy due to the side-effects). Dr. Walsh's surgical innovations and the use of PSA screening have turned the numbers around. Prostate cancer is second only to lung cancer in incidence and fatality to men, and smoking is the only reason lung cancer has the edge. So quit smoking and get your PSA tested, guys. (Women, quit smoking and get a mammogram.) There's enough death around without jumping into the volcano. --- Making Light: Open thread 118 ::: January 18, 2009, 10:25 AM Caroline@134: Wool-eating is very dangerous to your cats, because it can clog or tear their guts. I'm pretty sure I've read it also may be a symptom of a nutrition deficiency. There are links of possible helpfulness on Metafilter. A vet is a good first step, but you may have to find an animal behaviorist. A friend of mine had a poodle who repeatedly ate her bedding. The friend tried replacing the bedding with an extra-tough mat. The poodle still managed to eat it, but died. --- Making Light: Leavin' on a jet plane? A few handy links before you go ::: January 18, 2009, 12:23 PM Thanks for the info, abi, but.... Somehow the timing of your information suggests that it is related to flights to Washington, DC. And it's a damn good feeling to think we're going to have a president for a change. I just wish I could look forward to the extravaganza. I live 1/2 mile north of the White House, and my impulse is to bar the doors and stay inside Monday and Tuesday, and hope the city is there on Wednesday. It occurs to me I could get out of grinch mode and set up a water station at 16th and Q, if I could get someone to carry the jug back and forth for refills, since I'm not supposed to lift over 10 pounds for the next couple of weeks. At least I would feel useful. It's way too late to be an official volunteer. Maybe I should leave it to people who thought ahead. --- Making Light: Leavin' on a jet plane? A few handy links before you go ::: January 18, 2009, 04:13 PM Sorry to vent my siege mentality on you, the innocent. I'll take it as a sign that I need to take it easy and calm down. Thanks for the flying info and wise words. --- Making Light: The true history of the Bush years ::: January 19, 2009, 12:12 PM John Chu@39: Anyway, I remember listening to an NPR report where Bush supporters hope that his presidency will look better in retrospect (cf. Truman). That's a pretty low bar--Truman isn't in it. More like Nixon, but I'm not sure even Nixon is low enough to compete for unimpressive rehabilitation. --- Making Light: The true history of the Bush years ::: January 19, 2009, 12:31 PM Charlie@43: In the presence of corruption and evil, I tend to consider inefficacy a virtue. Unfortunately, still 23 hours to the beginning of the swearing-in ceremony. --- Making Light: Brooklyn grand jury service ::: January 19, 2009, 01:14 PM Patrick: Sifton...does seem to have ultimately enjoyed the experience more than I have. I think that happens looking back. I've never done grand jury duty, but I've known some grand jurors. I wish you calm for the week, and healing after. --- Making Light: The true history of the Bush years ::: January 19, 2009, 05:04 PM abi@62: Eloquent, but you surely know innocence doesn't come back. Don't throw away your tinfoil helmet. Jnthn drvb is doing us an important service--the tantrums of the vanquished remind us that they're still out there. --- Newsgroups: sprouts-theory From: Dan Hoey Date: Tue, 20 Jan 2009 14:04:05 -0500 Subject: Re: 17- Josh Purinton wrote: > On Tue, Jan 20, 2009 at 10:24 AM, wrote: >> 3+ is in the database, for example with ABC.}ABD.}CEF.}DEF.}]! or >> with ABCDEFGH.}ABCDEFGH.}]! > After discussing this with Jeff, I have a question. > Let G = ABCDEFGH.}ABCDEFGH.}]! I understand that Glop found that G = > 3+ = {*3}. However, according to Aunt Beast, this is not the case. I > would like to resolve the discrepancy. > Let G' = ABCD.}ABCEFG.}DEFG.}]!. Note that G' is an option of G. > If G = {*3}, then G' = *3. That's where you go wrong. G' = *[3/(2/3210)(2/321)21], where I write "/" instead of "+" in the compact Conway notation. That is to say, G'={{*3}, {{*2},*3,*2,*1,*0}, {{*2},*3,*2,*1}, *2, *1} G' is an option of G, but G' is not an option of the canonical form of G. Instead, G' is a reducible option. That is to say, we may treat G as if its only option is *3, because if anyone were to play G', the the opponent could immediately play to the {*3} option of G'. I haven't found out what position is the {*3} option of G' yet; my tools for examining the Lemoine-Viennot database are somewhat primitive. Dan --- Making Light: Online live video streams ::: January 22, 2009, 01:10 PM Ginger@155: "What split infinitive?" I nearly asked, but since GIMF, I found that Pinker is fingering ...the prohibition against \342EURoesplit verbs,\342EUR in which an adverb comes between an infinitive marker like \342EURoeto,\342EUR or an auxiliary like \342EURoewill,\342EUR and the main verb of the sentence. Okay, so it's an auxiliary split from its verb. Yesterday, a psychologist advanced the opinion that it comes from an attack of nerves coupled with the overwhelming nerve of not bringing a notecard, just in case. "Are you ready, Senator?" indeed. Barack had been president for several minutes already, oath or no oath. The next time Stevens administers the oath, I hope someone has the guts to ask, "Are you ready, Chief Justice?" --- Making Light: Online live video streams ::: January 22, 2009, 01:15 PM My niece went downtown Tuesday and got to see the Obamas walk on Pennsylvania Avenue. Woot! --- Newsgroups: sprouts-theory From: Dan Hoey Date: Thu, 22 Jan 2009 18:16:58 -0500 Subject: Re: Chained pivot/loop notation: a proposal "Josh Purinton" wrote to suggest that the neo-Glop-notation position 0.AB/0.0.0.AB.CD.EF/0.CD/0.0.EF be written in pivot/loop notation as 1L3(L1)(L2). I have a slightly different suggestion: that the position be written 1L3((L1))((L2)). Why the extra parentheses? Because then we can add Glop notation to pivot/loop notation by writing it inside single parentheses, AND we can add pivot/loop notation to Glop notation by writing it inside single parentheses. The double parenthesis notation is a simple consequence of the two abbreviations. So we have the well-known equality ...P0P0 = ...P1 . This could also be written ...P(2) = ...P(0). In this way the pivot/loop notation becomes completely as expressive as Glop notation. And we can write Glop positions like 0.0.0.0.0.0.AB/0.0.0.1<1>.2.AB as (6LA5(2)). By the way, today I proved the equality ...L(2) = ...L(0). This may be a much more useful equality, since it can cut down the size of a game tree right at the root. As with the other equality, it holds in both normal and misere play. Dan --- Newsgroups: sprouts-theory From: Dan Hoey Date: Thu, 22 Jan 2009 18:33:34 -0500 Subject: Re: Sprouts notation I propose using parentheses in this neo-Glop notation in a different way: Parentheses can enclose a pivot/loop chain as proposed by Josh Purinton in another sprouts-theory thread. If we add my other proposal, that parentheses be used in pivot/loop notation to enclose neo-Glop notation, we can use double sets of parentheses if neo-Glop needs grouping. I'm not sure about a good notation for repetition in neo-Glop, but pivot/loop notation makes most of it unnecessary. When parentheses are used in pivot/loop notation, I expect they introduce a new boundary in the region. In neo-Glop, parentheses may introduce part of a boundary, as in 0.2(L1)<(P2)> , which means 0.2AB/0.AB/0.0.C . --- Newsgroups: sprouts-theory From: Dan Hoey Date: Thu, 22 Jan 2009 20:34:06 -0500 Subject: A new equivalence I've already discussed a notation for partial positions such as ^xy:xy.0!, which refers to the pivot-loop fragment ...L0. Also, in the "Algorithms for planar graph canonization" thread I discussed some ideas on partial positions that lead to a proof that ^x:x.0! = ^x:x2! in normal or misere games. However, I am going have to extend my analysis somewhat to deal with partial positions that have more than one parameter. A _partial position (of order K)_ is a set of regions that share K pivots with other regions of the game. I will not now require that the pivots link to a single exterior region. If P is such a partial position, we call p1(P) the set of parameters of P, and p2(P) the set of (unordered) pairs of parameters of P. The interaction of P with the rest of the game is determined by the functions E*(P) and I*(P), defined below. E*(P) defines the result of moves exterior to P, and I*(P) defines the result of moves interior to P. Define E*(P) to be a function from p1(P) union p2(P) to the set of partial positions that arise from removing one or two parameters from P, as follows. For each pivot x in p1(P), E*(P)(x) = E_x(P) is the partial position of order K-1 formed by removing the parameter x from P. For each pair {x,y} in p2(P), E*(P)({x,y}) = E_xy(P) is the partial position of order K-2 formed by removing the parameter x and y from P. Define I*(P) to be a function from {{}} union p1(P) union p2(P) to the set of partial positions that arise from moving within P, as follows. I*(P)({}) = I(P) is the set of partial positions of order K that arise from moves not involving any parameters of P. For x in p1(P), I*(P)(x) = I_x(P) is the set of partial positions of order K-1 that arise from moves connecting the pivot x to a non-parameter of P. For {x,y} in p2(P), I*(P)({x,y}) = I_xy(P) is the set of partial positions of order K-2 that arise from connecting pivots x and y in a region of P. Now we define what it means for a partial position to be reducible to another. Suppose P and Q are partial positions of order K. I will consider that the parameters of P and Q are given the same names in the same order, though in practice we may need to use a bijection between the parameter sets. I will say that P and Q are coreducible if P'=Q', where P' is either P or a partial position to which P is reducible, and Q' is either Q or a partial position to which Q is reducible. Suppose in addition 1. Every E_x(P) and E_x(Q) are coreducible. 2. Every E_xy(P) and E_xy(Q) are coreducible. 3. Every element of I(Q) is coreducible with an element of I(P), and for every partial position S in I(P) not coreducible with an element of I(Q), I(S) has an element coreducible with Q. 4. Every element of I_a(Q) is coreducible with an element of I_a(P), and for every partial position S in I_a(P) not coreducible with an element of I_a(Q), I(S) has an element coreducible with Q. 5. Every element of I_ab(Q) is coreducible with an element of I_ab(P), and for every partial position S in I_ab(P) not coreducible with an element of I_ab(Q), I(S) has an element coreducible with Q. Then we say P is reducible to Q. Theorem: Suppose P, Q, and S are three partial positions of order K, such that P and Q have the same parameter set r...s, while S has the parameter set x...y. Then the positions PS=r...s=x....y! and QS=r...s=x...y! have the same outcome in normal play, and have the same outcome in misere play except possibly when QS=r...s=x...y! is the endgame. The proof is a straightforward exercise in Sprague-Grundy theory. The exception in misere play accounts for the proviso. [QED] I'll now describe an important result of the theorem, which may have immediate application in Sprouts play. I have already mentioned that ^x:x.0! is reducible to ^x:x2! . I haven't mentioned that ^x:x<1>! and ^x:x2.1! are also reducible to ^x:x2! . (It is notable that ^x:x12! is _not_ reducible to ^x:x2! . The nonequivalence can be demonstrated by joining the positions to ^x:xAB/A2/B2!.) The important result is that ^xy:xy.0! is reducible to ^xy:xy2! . The demonstration is as follows: 1. E_x(^xy:xy.0!) = ^x:x.0!; E_x(^xy:xy2!) = ^x:x2! . These positions have already been shown to be reducible. The demonstration for E_y is identical. 2. E_xy(^xy:xy.0!) = 0! = *[0]; E_xy(^xy:xy2!) = 2! = *[0] . 3. I(^xy:xy.0!) = {^xy:xy.AB/AB!}, and I(^xy:xy.AB/AB!) contains ^xy:xy2. The remainder of I(^xy:xy.0!) is {}, which is equal to I(^xy:xy2!). 4. I_x(^xy:xy.0!) = {^x:x<1>}; I_x(^xy:xy2!) = {^x:x2!} . These positions have already been shown to be reducible. The demonstration for E_y is identical. 5. I_xy(^xy:xy.0!) = 0.2! = *[1]; I_xy(^xy:xy2!) = 22! = *[1]. Therefore ^xy:xy.0! is reducible to ^xy:xy2! . [QED] This reducibility result allows us to shorten the game tree by two moves when L1 is a good move. Dan --- Making Light: Open thread 118 ::: January 22, 2009, 09:01 PM Raphael@266: Google "democracy patch" and ask for the cached version. I feel you will be lucky. --- Newsgroups: sprouts-theory From: Dan Hoey Date: Sat, 24 Jan 2009 09:54:15 -0500 Subject: Re: A new equivalence Yper Cube wrote: > I haven't thoroughly read all your posts about notation. Is this > ^xy:xy.0! is reducible to ^xy:xy2! > equivalent to the (GLOP) notation? : > }0.AB.} == }2AB.} Yes, that's what it means. Dan --- Newsgroups: sprouts-theory From: Dan Hoey Date: Sat, 24 Jan 2009 12:53:19 -0500 Subject: Re: A new equivalence Yper Cube wrote: > I can't remember if I had found this equivalence or if someone else (Jeff, > Josh?) had told me about it, but it was surely known long ago. It's good > that we now have a formal proof. I'm surprised. We all knew that }0.A.} == }2.A.} . Did you know that }2.A.} == }0.1aAa.} == }1.2A.} != }12A.} ? > But the proofs for bigger equivalences (with more liberties) may get > cumbersome in this notation. Reading this proof gives me haedaches to be > honest :) I agree it's long and tedious, but I want to make sure it's correct. In the end, I'm more interested in having a program do the proof. Dan --- Newsgroups: sprouts-theory From: Dan Hoey Date: Sat, 24 Jan 2009 18:24:09 -0500 Subject: Re: A new equivalence I wrote: > I'm surprised. We all knew that }0.A.} == }2.A.} . Did you know that > }2.A.} == }0.1aAa.} == }1.2A.} != }12A.} ? Sorry, I mistranscribed that. I meant to say }2.A.} == }1aAa.} = }1.2A.} != }12A.} Dan --- Newsgroups: sprouts-theory From: Dan Hoey Date: Sun, 25 Jan 2009 12:16:29 -0500 Subject: Re: A new equivalence At 10-Dec-08, ypercube replied: > I fear I've only scratched the surface of equivalences. I'm pretty > sure many more can be found. Even more, some of them, like the > (somehow escaped from being posted last time) }0.AB.} = }2AB.} can > have great impact on fast computing as it lowers the number of lives > by 2 (gain-2). I had looked at your Equivalences notes, and I'm very impressed, but I'm afraid I haven't gotten them down well enough that I recognize them all the time. So please excuse me if I fail to notice that you've sent them out first. One thing I don't recall seeing is the case where two partial positions differ by *[1]. The new misere Glop uses the fact that *[1]+*[1]=*[0] to gain two ply, so these may be useful in situations where there is another odd component: }1A.} = *[1] + }2A.} }AB.}BC.}1C.} = *[1] + }ABC.}BC.} The first is pretty obvious, but I don't know if the second has been noticed before. Dan --- Newsgroups: sprouts-theory From: Dan Hoey Date: Sun, 25 Jan 2009 16:18:20 -0500 Subject: Re: A new equivalence With regard to }1A.} = *[1] + }2A.} }AB.}BC.}1C.} = *[1] + }ABC.}BC.} Yper Cube wrote: > Both of these equivalences are just amazing! I think we need a name > for this (new to me) type. I don't think they are really different in kind. For instance, we could write them as }1A.} = }2A.}1.} }AB.}BC.}1C.} = }ABC.}BC.}1.} Dan --- Making Light: Ow ::: January 26, 2009, 05:24 PM Here's hoping you feel better soon, Patrick, both virally and moralely. When you get better and catch up with the overdue mountain I hope you feel better about having done the hard work of making the justice system work as well as it can. I know a lot of people who would make a grand jury worse, but I don't know anyone who I'd expect to do better. --- Newsgroups: sprouts-theory From: Dan Hoey Date: Thu, 29 Jan 2009 22:49:03 -0500 Subject: Re: Fwd: Sprouts equivalences I am disappointed by the failure of this equivalence, too. Let's consider two sequences of order-1 partial positions, P_n = ^x:x.((2))*n and Q_n = ^x:x2.((2))*n. In traditional Glop notation, that is P_0=}A.}, P_1=}A.2}, P_2=}A.2.2}, ..., and Q_0=}A2.}, Q_1=}A2.2.}, Q_2=}A2.2.2.}, .... As in my post on "A new equivalence", these partial positions X are determined by E_x(X), I(X), and I_x(X), where E_x(X) is the position that arises from using the pivot externally, I_x(X) is the set of positions arising from using the pivot internally, and I(X) is the set of partial positions that arising from moving internally without using the pivot. In tabular form, the results for the P_i are as follows. E_x I_x I P_0 *[0] {} {} P_1 *[0] {*[0]} {} P_2 *[1] {*[1]} {P_1} P_3 *[0] {*[0]} {P_2} Comparing P1 and P_3, we see the only difference is that P_2 is in I(P_3}, but since P_1 is in I{P_2} the P_2 is reversible. Therefore P_1 = P_3, and similarly P_n = P_(n+2) for n>0. Furthermore, E_x I_x I P_2+*[1] *[0] {*[0]} {P_1+*[1], P_2} which is equal to P_1, since both P_1+*[1] and P_2 reverse to P_1. This means that P_n + *[1] = P_(n+1) for n>0. Now consider the Q_i. E_x I_x I Q_0 *[0] {*[0]} {} Q_1 *[1] {*[1]} {Q_0} Q_2 *[0] {*[0],*[1]} {Q_1} Q_3 *[1] {*[0],*[1],*[2]} {Q_2} Q_4 *[0] {*[0],*[1],*[3]} {Q_3} Q_5 *[1] {*[0],*[1],*[2]} {Q_4} We would like to reverse Q_4 to Q_3 to make Q_5=Q_3. Unfortunately, I(Q_3) has the option Q_2, which I(Q_5) does not have. The same problem arises with any attempt to find equivalence among the Q_i. With *[1], E_x I_x I Q_0+*[1] *[1] {*[1]} {Q_0} Q_1+*[1] *[0] {*[0]} {Q_0+*[1], Q_1} Q_2+*[1] *[1] {*[0],*[1]} {Q_1+*[1], Q_2} Q_3+*[1] *[0] {*[0],*[1],*[3]} {Q_2+*[1], Q_3} Q_4+*[1] *[1] {*[0],*[1],*[2]} {Q_3+*[1], Q_4} we have that Q_0+*[1] = Q_1 (not surprising, since Q_0=P_1 and Q_1=P_2), but no other equivalences appear. I would be interested in seeing partial positions that can be used to distinguish Q_n, Q_(n+1)+*[1], and Q_(n+2). I am also interested in seeing whether a single partial position can be used for arbitrarily large n, or whether more and more complicated partial positions will be needed. Dan --- Newsgroups: sprouts-theory From: Dan Hoey Date: Fri, 30 Jan 2009 08:31:50 -0800 (PST) Subject: Re: A lousy periodicity theorem I wrote: > A "louse" is a boundary > consisting of a single degree-2 point that does not appear anywhere > else; "2." in Glop notation. We can change a sprouts game by adding > a louse to a region. The theorem is that beyond a certain number > (depending on the region), adding two lice to a region does not affect > the game. I regret having to withdraw this theorem. Given new observations in the "Fwd: Sprouts equivalences" thread, I see problems in my proposed method of proof that render this claim quite dubious. Dan --- Newsgroups: sprouts-theory From: Dan Hoey Date: Fri, 30 Jan 2009 11:22:18 -0500 Subject: Re: Fwd: Sprouts equivalences Yper Cube wrote: > We obviously don't have the same results for the Q_n. > But we do have this: > For any R, if there is a k>2 such as, R#(Q_k) + *1 = R#(Q_(k+1)) > then > for any n>=k, R#(Q_n) + *1 = R#(Q_(n+1)) > But this k is dependent on R so the equivalences are not "global" Thanks for making that explicit, Yper Cube. We can define lim_Q(R)=k in this case, and lim_Q(R)=infinity if there is no such k. Now I can see three possible situations for limset(Q) = { lim_Q(x) : x a partial position} 1. limset(Q) contains infinity. That means there is some R such that R#(Q_n) + *1 is never equal to R#(Q_(n+1)). This is the situation that you explain in your next message. 2. limset(Q) is an infinite set, but does not contain infinity. That would mean that there is a k as above for any R, but it may be arbitrarily large. 3. limset(Q) has a finite maximum K. This would mean that for k > K, Q_k + *1 = Q_(k+1) , even though we don't know how to prove it. I really don't know which of these is most likely to be the case. For any partial position S = }w.} where w is a string of symbols defining S, we can define S_1=}2.w.}, S_2=}2.2.w.}, .... and continue to define lim_S(R) and limset(S). The "lousy periodicity theorem" I described earlier is dependent on case 2 or 3 holding for every S . I have to withdraw that theorem; I'm not even willing to offer it as a conjecture any more. At least we now have a known unknown. Dan --- Newsgroups: sprouts-theory From: Dan Hoey Date: Fri, 30 Jan 2009 16:54:29 -0500 Subject: Re: Fwd: Sprouts equivalences I wrote: > Yper Cube wrote: >> We obviously don't have the same results for the Q_n. >> But we do have this: >> For any R, if there is a k>2 such as, R#(Q_k) + *1 = R#(Q_(k+1)) >> then >> for any n>=k, R#(Q_n) + *1 = R#(Q_(n+1)) >> But this k is dependent on R so the equivalences are not "global" > Thanks for making that explicit, Yper Cube. I should say more--thanks for showing that this is the case. It is a consequence of the particular sequence Q_k that if R#Q_k + *1 = R#Q_(k+1) for k>1, then R#Q_(k+1) + *1 = R#Q_(k+2), and therefore for all n>k. > We can define lim_Q(R)=k in this case, and lim_Q(R)=infinity if > there is no such k. [...] > limset(Q) = { lim_Q(x) : x a partial position} [...] > For any partial position S = }w.} where w is a string of symbols > defining S, we can define S_1=}2.w.}, S_2=}2.2.w.}, .... and > continue to define lim_S(R) and limset(S). Here I must be a little more careful, and define lim_S(R) to be the least k for which R#S_k + *1 = R#S_(k+1) = R#S_(k+2) + *1 , since for general partial positions S we cannot conclude the second equality from the first. Dan --- Making Light: Open thread 121 ::: March 24, 2009, 09:34 AM Linkmeister, #171 et al: You might want your calculator to graph f(x)=(sin x)/x. It's no matter what the graph shows for x=0, random value or gap or big red splotch. But what you want coming out of it is a graph, not an error message. For that matter, you might want to graph f(x)=1/x. Simple-minded cutoffs that omit the graph when it goes off the chart are easy, but you still don't want the machine to choke at x=0. --- Making Light: Open thread 121 ::: March 26, 2009, 12:19 PM abi@245: Thanks for that link. I was really impressed about how well they did the reverse doge hat haircut. In another clip, you see microphones in it, creating the appearance of a robotic hairpiece. And Patrick@244: I imagine you were referring to the noir atmospherics and schizophrenic geography of Mahagonny, but abi's video also contains a bizarrely coincidental homage in the form of a crowd of women with suitcases. --- Making Light: Open thread 121 ::: March 26, 2009, 02:09 PM KeithS@299: Why I didn't think microphone I have no idea. I cheated and read the youtube comments. Fortunately, there was one about the microphones in English. (Perhaps Dutch speakers had more interesting things to discuss than stagecraft.) --- Making Light: Open thread 121 ::: April 01, 2009, 05:12 PM Clifton Royston @496: Thanks for pointing out the gem of that page. It's certainly harder to skim that sort of discussion. --- Making Light: Open thread 122 ::: April 23, 2009, 12:40 PM Fragano@632: It's amazing what a homophone can do to two clich\303\251s, too. I ran into a mention of someone going "behind the veil of tears" a while back. --- Making Light: Dresden Codak ::: May 07, 2009, 12:59 PM Harriet@10.. Those links should be The Witching Hour, In the Company of Science, and She is the very model of a Singularitarian. You can also get there by way of the "Next" link on the previous pages. --- Making Light: Open thread 123 ::: May 07, 2009, 01:06 PM Fragano@391... I'm trying to imagine how someone could confuse Louis Antoine Juchereau de St. Denis with Christopher Columbus, or otherwise what the significance of 1714 might be. --- Making Light: Open thread 126 ::: June 25, 2009, 11:00 AM Terry Karney at 209: I suppose you put the cap on to prevent criticism of the resulting photography? --- Making Light: Dreadful phrases ::: July 31, 2009, 12:52 PM It's worth noting that linguistic reanalysis, also called an eggcorn, mondegreen, or pullet surprise, was also discussed in Open Thread 37. I'm still fond of the second-level reanalysis "a dog y dog world". --- Making Light: Life affords few such opportunities ::: August 06, 2009, 03:40 PM HA HA HA HA .... --- Making Light: Life affords few such opportunities ::: August 07, 2009, 12:06 PM 55: Verily it is said, "Who picks by the nit will be picked by the nit." I have it from reliable sources that Fitzgerald did not write "Bernice boobs her hair." My apologies for any confusion. --- Making Light: Open thread 128 ::: August 20, 2009, 12:31 PM Serge @545 The wikipedia article says that Peter/Pierre/Petra/Petros/\317EUR\316\265\317"\317\316\277\317' is a translation of the apostle's Aramaic name Kephas or Cephas, all meaning "stone" or "rock". --- Making Light: All Knowledge Is Contained ::: September 01, 2009, 08:45 AM Zillions of hits today on "tell me right I wrote the following sentence" and "give true I wrote the following sentence". Someone is ESL-bombing the internets. I suspect the payload is in the header URL. Though what "Members/ Thinning" is supposed to represent is anyone's guess. Hair club for morons? --- Making Light: Flash of insight: swift, blinding, pointless ::: September 01, 2009, 12:27 PM Lenny Bailes @36: Your link gets me to a Google groups "topic not found" page for rec.arts.sf.fandom. Tinyurl considered harmful. --- Newsgroups: rec.puzzles From: Dan Hoey Date: Thu, 10 Sep 2009 22:30:21 -0400 Subject: Re: Sqzuaring the Circle Mike Williams wrote: > Wasn't it Onoit who wrote: >> It seems to me it would be easy to square a circle. Just wrap a >> line of string around a jar lid to form a circle, cut the string >> and glue the ends together. Form a square of 4 thumb tacks, and >> wrap the string around the thumbtacks. Resize the square until the >> string is taut. Would this work? > Many of the classic impossible compass and straight-edge > constructions are quite possible if you permit the use of other > equipment. However, in this particular case the "resize the square" > bit would need to be made more rigorous before I'd consider it > valid. [...] Yes, but in this case resizing the square is the easy part. The part most unlike classic construction is wrapping the string around the jar lid. If you can wrap the string around the jar lid, lay the string along a line on the paper and mark the length of the string as the segment AB of the line. Then mark three points from the jar lid on your paper, and construct the center of the circle through those three points. From the center to any one of the points is the radius R of the jar lid. Draw the circle of radius R and center A. Let C be the intersection point furthest from B of the circle with the line, so AC=R. Bisect AB, and let the midpoint be D, so AD=Pi R. Now erect a perpendicular to CD at A, and construct a circle with diameter CD. Let H be an intersection point of the perpendicular and circle. Then AH is the geometric mean of AC and AD, and a square with side AH has area Pi R^2 Q.E.C. Dan Hoey haoyuep at aol.com --- Newsgroups: rec.puzzles From: Dan Hoey Date: Thu, 10 Sep 2009 22:39:51 -0400 Subject: Re: Two numbers Mark Brader wrote: > Jonnie Grint: >>> I can't find any other similar pairs of numbers, but your puzzle >>> reminds me of a verse... > Mark Brader: >> Correct! > I guess it's time to explain this. > The two numbers aren't really equal to each other, but they are both > equal to DIX. One as a word in French, the other as a Roman numeral. > I noticed this when 509 = DIX came up in the other number puzzle > I posted recently. We might also say that 1049 = 0.001 in a similar sense that works in American. Obpuzzle: What integer is sqrt(x)? Dan --- Making Light: Open thread 129 ::: September 11, 2009, 02:13 PM Bruce Cohen #516: On the question of Penrose tiles, non-periodic tilings are not difficult or new--you can make a non-periodic tiling from copies of a single triangle tile.* What has been developed since 1961 are aperiodic tilings, tilings of the plane with tile shapes that admit no periodic tiling. The first aperiodic tilings were due to Hao Wang using large numbers of tile shapes based on squares. Penrose was the first to demonstrate aperiodic tilings with only two shapes of tile. Penrose used shapes based on rhombuses, kites, and darts with pentagonal symmetry, and these shapes appear in classical Moorish tilings. However, these shapes do not by themselves generate aperiodic tilings, because the tile shapes can be used to tile periodically. It is only the particular matching constraints introduced by Penrose that make the tiling aperiodic. So the Moorish tilings are not the same as "Penrose, or quasi-crystal tesselations"\342EUR any more than a tiling by squares is a Wang tiling. Aperiodic tilings from a finite set of tile shapes were not demonstrated before Wang in 1961. The interest in aperiodic tilings is implied, though, as early as Hilbert's eighteenth problem from the last year of the nineteenth century. *For instance, an isosceles right triangle can be used to tile the plane periodically as squares bisected along a diagonal. A subset of the squares may be modified to use the other diagonal, producing a non-periodic tiling. \342EUR Though a large amount of confusion between the two is available on the net. The nature of Penrose's discovery is subtle and prone to misinterpretation, particularly if one concentrates on diagrams that do not emphasize the aperiodicity constraints. --- Making Light: Open thread 129 ::: September 11, 2009, 05:38 PM Bruce Cohen @ 694: I think you need to reread #595. The Moorish tilings based on kites and darts are not aperiodic. Kites and darts can be assembled to form periodic tilings. It is necessary to add constraints, as Penrose did, to get aperiodicity. --- Making Light: Open thread 129 ::: September 11, 2009, 10:19 PM This discussion led me to review the Wikipedia article on Penrose tiles, and I see that Penrose acknowledged Kepler's work on pentagonal tilings in Harmonices Mundi as the inspiration for his development of pentagonally-based tiles. Of course, Kepler's tilings were not aperiodic any more than the Moorish ones. Perhaps one reason for Penrose's interest is the well-known result that a tiling with fivefold rotational symmetry about a point cannot be periodic. Thus if a set of tile shapes could enforce fivefold symmetry, the tiling would be aperiodic. While Penrose did find a way to ensure aperiodicity in pentagonally-based tiles, it is notable that his tiles do not enforce fivefold symmetry. In fact, only two out of the 2^\342"\265[0] Penrose tilings possess fivefold symmetry. --- Newsgroups: rec.puzzles From: Dan Hoey Date: Sun, 13 Sep 2009 00:58:40 -0400 Subject: Re: Two numbers Ed Murphy wrote: > Dan Hoey wrote: >> Obpuzzle: What integer is sqrt(x)? > Presumably V. I had in mind III, since the French word for "x" is IX. --- Making Light: Open thread 129 ::: September 14, 2009, 01:03 PM Bruce Cohen @ 604: Please disregard my response at #608. I certainly should have considered that the news was supposed to be surprising, and so counter to prevailing belief (at least, prevailing up to four years ago). In or out of context, though, I believe my response was arrogant and rude to you, and I deeply regret it. I once again resolve not to act that way in the future. I've looked at the article (which is now up on the Saudi Aramco World site) and the Science article on which it is based, and I am quite astonished to see that some 12-15th century artisans apparently developed the concept of hierarchical tiling (as described in Wikipedia under "Aperiodic tiling"). As far as I can tell, these tiling methods actually do fall short of being aperiodic, in that there is no development of local adjacency constraints to enforce the hierarchical structure of the tiling. However, the use of hierarchical tiling is itself quite amazing. Thank you for bringing this to the fluorosphere. --- Making Light: The Bully Pulpit ::: September 14, 2009, 10:38 PM As a somewhat socially inept person (whose self-image is even moreso) I would normally have absolutely nothing to say on this topic. I would probably have given up reading within the first ten comments, if I even made it to the cutline, on the grounds that this discussion is in some dialect that I've never been able to decipher. But by some accident of synchronicity, I've been examining my own behavior on ML, due to a suspicion that one of my recent comments was not terribly appropriate. So I decided to brave the incomprehensibility in hopes of understanding a little of what sort of posting is being deprecated here. I've had two or three warnings from PNH in years past about the "huffy" tone I sometimes assume. I never really knew what he meant by that, except that the objectionable comments were ones in which I had to admit I wasn't thinking too thoroughly about what I was saying and how I was saying it. This topic, though, provides a description of an academic/authoritarian bully that is an embarrassingly good match for me when I'm being huffy. It has helped me understand both the insecurity that triggers the tone and the poisonous results of using it. Most importantly, it gives me a more explicit description of what kind of writing I need to guard myself against using. With this I have a chance at catching myself while I'm writing, rather than after I've posted. So thanks, abi, for leading us to this discussion. Thanks, commenters, for the insights that have made it through my perceived- social-incompetence filters. And thanks to the whole fluorosphere for having put up with my occasionally semi-toxic writing pending the latest object lesson, which I sincerely intend to be the last of its kind. I look forward to repaying your patience. I still don't find this stuff easy reading, but I guess learning stuff is hard. --- Making Light: Open thread 125 ::: September 15, 2009, 08:42 AM The text of the spam is too good to lose: I have noticed many changes in your blog and they are like improvements for you. Who needs fortune cookies when we have blogspam? --- Making Light: Open thread 129 ::: September 15, 2009, 10:12 AM Cliff Royston @ 662: Note that the eigenvalue-based proof only shows that hierarchical tilings (using a particular dissection rule) must be non-periodic. The tiles themselves admit periodic tilings. So there is no aperiodicity to be found there. --- Making Light: Open thread 130 ::: October 09, 2009, 11:24 PM Poetry should be italicized when it consists of mathematical variables, titles of books and periodicals, and emphasized words; there are probably other cases that I'm not recalling at the moment. This being the fluorosphere, I have some hope that someone will take that as a challenge. --- Wikipedia User talk:Dominus/Archive (2009) Parasitic Numbers vs. Palintiples Actually, I don't believe I did anything on Parasitic numbers. I think you're remembering Palintiples, numbers that can be multiplied by an integer by reversing their digits. If you consider them encyclopedic, go boldly. I suspect that as nearly* the only source of original research on the subject, it would be in poor taste for me to edit on the subject, but GIYF. (*)There's some intriguing related work by L. A. Graham, probably in The Surprise Attack in Mathematical Puzzles. I suppose I could edit on that if desired. -Dan Hoey 01:26, 23 October 2009 (UTC) --- Wikipedia Talk:The Legend of Neil Wikimarkup fail Anyone know why the wikitable of Season 1 episodes appears at the end of the article? -Dan Hoey 03:33, 23 October 2009 (UTC) --- Making Light: Open thread 130 ::: October 23, 2009, 11:31 AM P J Evans @820: Beautiful pictures! And so reminiscent of canals. I'm trying to imagine how there could be a related phenomenon on a much larger scale. --- Wikipedia Talk:The Legend of Neil Wikimarkup fail Answer: the wikitable was improperly terminated. -Dan Hoey 04:54, 5 November 2009 (UTC) --- Making Light: Open thread 132 ::: November 27, 2009, 05:48 PM AKICIML: In an NPR interview, Daniyal Mueenuddin likens Palestine's madrassas to The Lord of the Rings, in which (he says) there's a wonderful scene in that in which the evil lord, whose name I forget, is sort of building an army out of clay and dust and underground, these vats, making these evil warriors. Does everyone agree that he's got the wrong book? Anyone care to venture a guess what work he's actually remembering? --- Making Light: Open thread 132 ::: November 27, 2009, 11:57 PM Thanks, Lizzy L (386) and KeithS (387) for pointing to the Uruk-hai in the movie. --- ErfWiki User:Dan Hoey Dan Hoey, also a wikipedia editor. This page was last modified on 5 December 2009, at 02:53. --- ErfWiki Talk:Carl Is that an "I/O" cap? It looks like the IEC 5009 symbol for sleep mode (or simply "power" according to IEEE 1621). -Dan Hoey 03:28, 5 December 2009 (UTC) --- ErfWiki Talk:Carl I got the better description from TBFGK_15:7 referring to the 'international "on/off" 1/0 symbol'. I don't know if it's worthwhile getting into "sleep" mode. -Dan Hoey 07:34, 5 December 2009 (UTC) --- ErfWiki User talk:Dan Hoey Editing annotations, especially footnotes Up to TBFGK_86 so far. -Dan Hoey 13:51, 5 December 2009 (UTC) --- ErfWiki User talk:Dan Hoey Talk page of Dan Hoey, also a wikipedia editor. Until I get erfwiki e-mail working, I suggest you use my WP talk page for comments. Yay! If I count for experienced, we're all in trouble. But I don't know what you're asking. Did you want me to put {{mainpagecontrol}} on my home page, talk page, or where? Where do I get the data to fill in all the empty slots? -Dan Hoey 13:55, 5 December 2009 (UTC) --- ErfWiki User talk:Dan Hoey Editing annotations, especially footnotes Why the booping boop doesn't the talk link work? -Dan Hoey 00:42, 6 December 2009 (UTC) --- ErfWiki User talk:Dan Hoey Editing annotations, especially footnotes Something that works everywhere except User_talk: pages? There's gotta be a technical reason for this, but it's still ridonkulatious. -Dan Hoey 00:48, 6 December 2009 (UTC) --- ErfWiki User talk:Dan Hoey Editing annotations, especially footnotes Up to TBFGK 100 so far -Dan Hoey 03:03, 6 December 2009 (UTC) --- ErfWiki User talk:Commander I. Heartly Noah Footnotes on annotation pages Hi, I'm new here, and I noticed you added a footnote on TBFGK 59 for panel 2. When you edit the appropriate text, it puts the {{ref}} on both the page-by-page annotation TBFGK 59 and the panel-by-panel annotation TBFGK 59:2. But the {{note}} doesn't get duplicated. This means that people browsing panel-by-panel get a footnote marker that doesn't go anywhere. The way that I know works is to add the same {{note}} to both annotation pages TBFGK 59 and TBFGK 59:2. I don't know if this can be automated; it's a pain in the absence of automation, which I don't know how to do. But if it's not too much work, it will make the Annotation pages less confusing for readers. Or if you know of another way of preventing dead footnotes, I'm all ears! -Dan Hoey 03:25, 6 December 2009 (UTC) --- ErfWiki User talk:SteveMB Annotations references need a new standard In my Newbie NSHO: The references standard isn't really working, since folks don't pay attention to the standard. And even if they do, they can make mistakes that don't get fixed. I don't know how to fix that. I've gone through 100 pages of annotations, and found dozens of cases of missing panel annotation references, and page/panel annotation skew (when one footnote is updated but not copied to the other). I've also probably overlooked several cases. It's an impossible task. If there were some sort of subpage for the panel that had the footnote, the footnote subpages could be glommed together in the same way the panel text is done. But then we'd need to have a macro that showed the =====Notes===== section only when there was some text in the footnotes. That's wikitude way above my pay grade. If there were some sort of bot that scanned the Annotations pages for differences between the page-by-page and panel-by-panel notes, it might at least be feasible to repair the damage that we continually inflict. Again, I have no botmancy. Anyway, this is raising the issue, in case you or anyone else has the chops to attack it. -Dan Hoey 03:46, 6 December 2009 (UTC) --- ErfWiki User talk:Dan Hoey Editing annotations, especially footnotes To TBFGK 112, another page with numeric footnote tags. -Dan Hoey 03:28, 7 December 2009 (UTC) --- ErfWiki User talk:Dan Hoey Editing annotations, especially footnotes Thanks for the explanation-Dan Hoey 03:42, 7 December 2009 (UTC) --- ErfWiki User talk:Dan Hoey References on Annotations pages User talk:SteveMB gives a standard form for TBFGK and therefore other Annotations pages. I wrote some of this there, but it's here so I can look at it more. If someone else enters this discussion, we can put it in one place. The problem is that the standard is hard to follow, and does not prompt one to follow it. If someone adds a footnote to the page-by-page without adding the footnote to the panel-by-panel, it looks fine on the page-by-page. But the panel-by-panel has a footnote ref that doesn't have a footnote note. Similarly, someone can add a footnote to the panel-by-panel without adding the note to the page-by-page. Again it looks fine on the panel-by-panel but the page-by-page is screwed up. -Dan Hoey 03:28, 8 December 2009 (UTC) --- ErfWiki Template talk:Erf Link to Annotations? I'd find it more useful if the link was to the wiki Annotations page, rather than the archive page. Anyone else think so? -Dan Hoey 03:37, 8 December 2009 (UTC) --- ErfWiki Template talk:Blog Summer Updates Breaking Links Should there be a way to specify the appearance of the text and mouseover independent of the actual link, so we don't have links that look like ^summer-updates-%e2%80%93-046 (see Charlescomm). I mean, if we could say that the it should look more like ^Blog Entry summer- updates 046 with some more readable mouseover text. So it's not a big "if" statement, but the citer is responsible for saying what the cite should look like (if different from some default appearance). -Dan Hoey 03:48, 8 December 2009 (UTC) --- ErfWiki User talk:Dan Hoey References on Annotations pages The page-by-page is scaled down so you can't read it, and the relevant notes are too far away. But in panel-by-panel, you can see the whole thing, full size, in one screen. That's what does it for me. -Dan Hoey 01:20, 9 December 2009 (UTC) --- ErfWiki User talk:Dan Hoey References on Annotations pages By the way, it would be good to be able to make "random page" skip the panel-by-panel pages. -Dan Hoey 22:03, 12 December 2009 (UTC) --- ErfWiki User talk:Dan Hoey Editing annotations, especially footnotes Done TBFGK 121 -Dan Hoey 22:38, 12 December 2009 (UTC) --- ErfWiki User talk:Dan Hoey Editing annotations, especially footnotes At First Intermission 45 -Dan Hoey 21:41, 13 December 2009 (UTC) --- ErfWiki User talk:Dan Hoey Editing annotations, especially footnotes Now at First Intermission 47 -Dan Hoey 23:23, 13 December 2009 (UTC) --- Newsgroups: rec.puzzles From: Dan Hoey Date: Tue, 15 Dec 2009 15:00:33 -0500 Subject: Re: Integer right triangles Mark Brader wrote: > Rich Grise: >> Well, that got me to thinking - everybody knows about the 3,4,5 and the >> 5,12,13 right triangles; I'm wondering if, other than their multiples >> (6,8,10; 10,24,26 etc.) are there any other integer right triangles? > Yes. >> If so, is(are?) there an infinite number of them? > Yes. See e.g. http://mathworld.wolfram.com/PythagoreanTriple.html That page has a lot of stuff, making it easy to miss equation 11: If u > v are positive integers, relatively prime and not both odd, then the numbers 2uv, u^2-v^2, u^2+v^2 form a Pythagorean triple, and every Pythagorean triple is generated by exactly one such pair (u,v). Dan --- ErfWiki ErfWiki:Questions E-mail verification I didn't get any e-mail to confirm at my @aol address. I also tried my @gmail.com address and nothing arrived. Is this feature supposed to be working? -Dan Hoey 22:12, 21 December 2009 (UTC) --- ErfWiki ErfWiki:Questions Robotics I've been considering ways of checking consistency, e.g. that the panel-by-panel annotations have the same footnotes as the page-by-page footnotes. To do this right, I'd want to download wikitext of the annotations pages. Is there a way to do this? -Dan Hoey 22:12, 21 December 2009 (UTC) --- ErfWiki ErfWiki:Questions E-mail verification Oops, I had not tried the gmail.com address, and it worked. So maybe the warning about @hotmail should also mention @aol.com -Dan Hoey 22:15, 21 December 2009 (UTC) --- Making Light: Bruce Schneier on Rachel Maddow ::: December 29, 2009, 09:32 AM Washington, D.C.'s subway system lost its litter bins shortly after 9/11, but redesigned bins were installed thereafter. I don't know how the new ones are supposed to be safer than the old ones. But I'd rather not ask searching security questions, lest someone decide they aren't that safe after all and take them away again. ---