## Tuesday, December 31

## Sunday, November 10

### Define, divide, and conquer

I had done a table of smallest positive integer whose name has a given number (3-758) of letters and was anxious to tackle the largest-number analogue. The initial terms had a modicum of fame by being part of the Google Labs Aptitude Test (second-last item). I checked the sequence in the OEIS where it sported nine terms (3-11 letters) and the opinion that "beyond this point the terms are too ill-defined to include". So the first thing I did was provide a more tractable outcome for the sequence, accomplished simply enough by adding "less than 10^66" to the sequence definition.

I realize that a largest-number sequence limited in scope by a hard cut-off like this is somehow less natural (if words in any language could ever be considered

Once I did set about expanding the b-file, things just fell into place: terms for 631-701 letters were derived by brute force on my computer. Manually, I derived the terms for 757-702 letters (backwards). Then I had my computer calculate sections of 28 terms each surrounding (10^3n-1)*10^(66-3n) for an appropriate range of n. This gave me terms for 50-77, 84-111, 120-147, 153-180, 187-214, 225-252, 258-285, 291-318, 323-350, 353-380, 383-410, 413-440, 444-471, 475-502, 507-534, 539-566, and 568-630 letters.

These sections provided the anchors for what remained. Considering endings of 10*10^9 provided the terms for 43, 78, 112, 148, 181, 215, 253, 286, 319, 351, 381, 411, 441, 472, 503, 535, and 567 letters; endings of 10*10^12: 44, 79, 113, 149, 182, 216, 254, 287, 320, 352, 382, 412, 442, 473, 504, and 536 letters; endings of 10*10^33: 45, 80, 114, 150, 183, 217, 255, 288, and 321 letters; endings of 10*10^24: 46, 81, 115, 151, 184, 218, 256, 289, and 322 letters; endings of 10*10^36: 47, 82, 116, 152, 185, 219, 257, and 290 letters.

The remainder were done piecemeal, in this order: 48, 83, 117, 186, and 220 letters; 49, 118, and 221 letters; 119 and 222 letters; 223 letters; 224 letters; 443, 474, 505, and 537 letters; and finally, 506 and 538 letters.

After all that, I removed the keyword "hard" from this OEIS sequence. My table is here.

I realize that a largest-number sequence limited in scope by a hard cut-off like this is somehow less natural (if words in any language could ever be considered

*natural*) but it does no harm to the initial terms and provides a firm framework for many more. I derived the terms for 12-42 letters manually, more or less by inspection. I contributed the terms thus far as a b-file and noted the final (758 letters) number which I had previously encountered as a contribution from Eric Brahinsky on Jeff Miller's Word Oddities site, though since removed. Finally, I added the keyword "hard" because I did not see my way to solving for a lot more terms. Well, I hadn't at that point*tried*.Once I did set about expanding the b-file, things just fell into place: terms for 631-701 letters were derived by brute force on my computer. Manually, I derived the terms for 757-702 letters (backwards). Then I had my computer calculate sections of 28 terms each surrounding (10^3n-1)*10^(66-3n) for an appropriate range of n. This gave me terms for 50-77, 84-111, 120-147, 153-180, 187-214, 225-252, 258-285, 291-318, 323-350, 353-380, 383-410, 413-440, 444-471, 475-502, 507-534, 539-566, and 568-630 letters.

These sections provided the anchors for what remained. Considering endings of 10*10^9 provided the terms for 43, 78, 112, 148, 181, 215, 253, 286, 319, 351, 381, 411, 441, 472, 503, 535, and 567 letters; endings of 10*10^12: 44, 79, 113, 149, 182, 216, 254, 287, 320, 352, 382, 412, 442, 473, 504, and 536 letters; endings of 10*10^33: 45, 80, 114, 150, 183, 217, 255, 288, and 321 letters; endings of 10*10^24: 46, 81, 115, 151, 184, 218, 256, 289, and 322 letters; endings of 10*10^36: 47, 82, 116, 152, 185, 219, 257, and 290 letters.

The remainder were done piecemeal, in this order: 48, 83, 117, 186, and 220 letters; 49, 118, and 221 letters; 119 and 222 letters; 223 letters; 224 letters; 443, 474, 505, and 537 letters; and finally, 506 and 538 letters.

After all that, I removed the keyword "hard" from this OEIS sequence. My table is here.

## Saturday, August 31

### The 1%

This propaganda blurb is courtesy the Toronto Hydro-Electric System, accompanying my most recent every-other-month bill. The 1% struck me as unusually low. Looking over the source document, the relevant fact should be gleanable from the last column of this table, but (alas) electricity is grouped with water and fuel, which (in total) constitute 3.1% of the average 2009 household expenditure, the latest available. Why use the Canada-wide table instead of the Ontario one? Because Ontario's is slightly higher at 3.3%. How does Toronto Hydro arrive at 1%? Presumably by suggesting that electricity is

*one third*of the water-fuel-electricity expenditure. In*our*household (which uses natural gas for cooking and heating), electricity actually represents*half*of that water-fuel-electricity total.
Our entire 2009 income was just 46% of that average household expenditure suggested by Statistics Canada. We didn't, but suppose we spent every penny of our earnings that year. What fraction would have been for electricity? The answer is 3.7%.

## Sunday, August 25

### Square Split Subtract

There are

Square: 3636363636363636365

Split: (here, into two 19-digit parts) 1322314049586776860 ' 4958677685950413225

Subtract: 4958677685950413225 - 1322314049586776860 = 3636363636363636365

What are the other two solutions?

*three*(base-ten) 38-digit squares that can be split (somewhere) in such a way that (the difference between the two parts)^{2}is the original number. One of them is the square of 3636363636363636365:Square: 3636363636363636365

^{2}= 13223140495867768604958677685950413225Split: (here, into two 19-digit parts) 1322314049586776860 ' 4958677685950413225

Subtract: 4958677685950413225 - 1322314049586776860 = 3636363636363636365

What are the other two solutions?

## Saturday, August 3

### Cooking the books

## Tuesday, July 9

### Do without

*"Why shouldn't I buy it? I've got the money!"*

Sure you've got the money. So have lots of us. And yesterday it was all ours, to spend as we darn well pleased. But not today. Today it isn't ours alone.

*"What do you mean, it isn't mine?"*

It isn't yours to spend as you like. None of us can spend as we like today. Not if we want prices to stay down. There just aren't as many things to buy as there are dollars to spend. If we all start scrambling to buy everything in sight, prices can kite to hell-'n'-gone.

*"You think I can really keep prices down?"*

If you don't, who will? Uncle Sam can't do it alone. Every time you refuse to buy something you don't need, every time you refuse to pay more than the ceiling price, every time you shun a black market, you're helping to keep prices down.

*"But I thought the government put a ceiling on prices."*

You're right, a price ceiling for your protection. And it's up to you to pay no more than the ceiling price. If you do, you're party to a black market deal. And black markets not only boost prices — they cause shortages.

*"Doesn't rationing take care of shortages?"*

Your ration coupons will — if you use them wisely. Don't spend them unless you have to. Your ration book merely sets a limit on your purchases. Every coupon you don't use today means that much more for you — and everybody else — to share tomorrow.

*"Then what do you want me to do with my money?"*

Save it! Put it in the bank! Put it in life insurance! Pay off old debts and don't make new ones. Buy and hold War Bonds. Then your money can't force prices up. But it can speed the winning of the war. It can build a prosperous nation for you, your children, and our soldiers, who deserve a stable America to come home to. Keep your dollars out of circulation and they'll keep prices down. The government is helping — with taxes.

*"Now wait! How do taxes help keep prices down?"*

We've got to pay for this war sooner or later. It's easier and cheaper to pay as we go. And it's better to pay more taxes NOW — while we've got the extra money to do it. Every dollar put into taxes means a dollar less to boost prices. So...

Use it up ... Wear it out ... Make it do ... Or do without

[This advertising appeared in a number of U.S. magazines in early 1944.]

## Sunday, June 9

### Lore of the (complicated) rings

A puzzle such as the one illustrated was mentioned by Italian mathematician Luca Pacioli in his

*De Viribus Quantitatis*(1496-1508). (I should point out that the caption underneath the illustrated text at this site belongs underneath the rings picture

*to its right*.) Not long after Pacioli's mention, there is an apparent reference to it by Yang Shen (1488-1559): "Nowadays, we also have an object called nine linked rings. It’s made of brass or iron instead of jade. It’s a toy for women and children." The quotation is taken from this page. There have been attempts to place the puzzle in China even earlier: The oft-repeated story that it was invented by Hung Ming (181-234) belongs to Stewart Culin's

*Korean Games*(1895) and is clearly not a serious possibility. Another suggestion that it was known in the Sung Dynasty (960-1279) belongs, I think, to Ch’ung‑En Yü's Chinese

*Ingenious Ring Puzzle Book*(1958) or, more properly, to Yenna Wu's 1981 translation of it (neither of which I have seen), although V. Frederick Rickey suggests (perhaps erroneously) in this 2005 paper that it is from Stewart Culin. My take is that the Sung Dynasty connection is not, at present, credible.

Another European mention of the puzzle rings comes from Gerolamo Cardano's Latin

*De Subtilitate*(~1550). John Wallis gives us a thorough description and illustration of the 'complicated rings', as well as its solution, in his Latin*Algebra*(1693). We find another picture of the rings in Jacques Ozanam's Récréations Mathématiques et Physiques (~1723). Hung Lou Meng's*The Dream of the Red Chamber*(~1750) mentions the "nine strung rings" puzzle (in H. Bencraft Joly's 1891 translation). From Johann Nikolaus Martius, by way of Johann Christian Wiegleb, we have*Unterricht in der natürlichen Magie*(1789), where it is chaptered*Die Zauberkette oder das magische Ringspiel*, complete with solution. Zhu Xiang Zhuren's*Little Wisdoms*appeared ~1821. Concurrently, William Clarke's treatment in*The Boy's Own Book*(1828) is shown here in an 1849 edition. He mentions the puzzle being called*The Tiring Irons*. The article was reprinted without provenance in*The Magician's Own Book*(1857), which is identical to*The Book of 500 Curious Puzzles*(1859).
A significant treatment of the puzzle arrived with a pamphlet written by Louis Agathon Gros:

*Théorie du Baguenodier*(1872). David Singmaster was looking for this many years ago and I have not been able to find a copy either! Andreas M. Hinz, Sandi Klavžar, Uroš Milutinović, and Ciril Petr have recently named OEIS sequence A001511 the*Gros sequence*for this pamphlet's contribution.
A flurry of renewed interest rounds out the 1800s: The puzzle is in T. (which stands for

*what*?) de Moulidars'*Grande Encyclopédie des Jeux*(1888), Le baguenaudier; Édouard Lucas'*Récréations Mathématiques*(1891), Le Jeu du Baguenaudier; W.W. Rouse Ball's*Mathematical Recreations*(1892), Chinese Rings; and Professor Hoffmann's*Puzzles Old and New*(1893), Cardan's Rings. Lucas footnotes (via O.-J. Broch) that in Norway the puzzle was used as a lock, a subsequently much-touted fiction. Rouse Ball (according to Singmaster) already noted that "It is said — though a priori the fact would have seemed very improbable — that Chinese rings are used in Norway to fasten the lids of boxes, ... I have never seen them employed for such purposes in any part of the country in which I have travelled." The objection, alas, was dropped from the third edition of*Mathematical Recreations*. Rouse Ball may have been the first to reference Cardan as*paragraph 2*of Book 15, but I wonder if he was ignoring (or labeling as paragraph*zero*) the start of the*De Subtilitate*page (look for "Hoc instrumento ludus excogitatus mirae subtilitatis" near the bottom). Hoffmann talks of "the puzzling rings" and "the tiring irons" but makes 'Baguenaudier' feminine. He credits his solution to "an anonymous American writer" but it is clearly that of Clarke's*Boys Own*. Furthermore, his reference to another explanation in the*Encyclopédie Méthodique des Jeux*is likely meant to refer to that of Moulidars'*Grande Encyclopédie*.
Henry Ernest Dudeney has the puzzle as

*The Tiring Irons*in*Amusements in Mathematics*(1917). Singmaster has noted that "the OED entry at Tiring-irons gives 5 quotations from the 17C: 1601, 1627, 1661, 1675, 1690." He also notes the variants Tyring or Tarrying Irons, and Tarriours. From Culin (1895) we have "*Ryou-kaik-tjyo*(Chinese,*lau kák ch’á*), or 'Delay guest instrument', is the name given to the familiar ring and bar puzzle which the Chinese call*kau tsz' lin wán*, or 'nine connected rings'." Pieter van Delft and Jack Botermans call it*meleda*in*Creative Puzzles of the World*(1978). More specifically: "Meleda first appeared in Europe in the mid-16th century and was described by the Italian mathematician Geronimo Cardano." Some readers of this sentence seem to have misinterpreted it to mean that the*word*goes back to Cardan. In fact, the introduction of*meleda*into English (it appears already in the puzzle sense in an 1835 Russian-French dictionary) is likely connected to the 1963 Halina Moss translation of Aleksandr Petrovich Domoryad's 1961 Russian*Mathematical Games and Pastimes*. In*Puzzles Old & New*(1986), Jerry Slocum and Jack Botermans go back to calling the puzzle*Chinese Rings*and otherwise repeat a lot of questionable gossip (the "Seal of Salomon" almost certainly refers to a different puzzle).
Martin Gardner's August 1972 column in Scientific American,

*The curious properties of the Gray code and how it can be used to solve puzzles*, mentioned the Chinese rings. The article received a significant addendum in its reprinting as*The Binary Gray Code*in*Knotted Doughnuts*(1986) which mentions Sydney N. Afriat's*The Ring of Linked Rings*(1982), a book I have yet to read. Gardner suggested: "The Japanese became so intrigued by the puzzle in the 17th century that they wrote Haiku poems about it, and symbols of the linked rings appeared on heraldic emblems." These assertions beg for verification. Gardner also quotes the Oxford English Dictionary for this 1782 doggerel:*Have you not known a small machine*

*Which brazen rings environ,*

*In many a country chimney seen*

*Y-clep’d a tarring-iron?*

There is a more complete version (transcript) attributed to S.S. (

*attributable*to William Shenstone) in*The Gentleman's Magazine*Volume X (October 1740). The mention of the puzzle in chimneys is surely a bit of a mystery. Dudeney (1917) said that "it is said still to be found in obscure English villages (sometimes deposited in strange places, such as a church belfry)." What's up with that?## Sunday, May 19

### A Canadian propensity for environmental hyperbole

I was more than a little confused when the (magnitude 4.4) May 17 Shawville earthquake was persistently described by the media as being magnitude 5.2. Canadians are totally insecure about their place in the world so, when it comes to reporting the environment, bigger is better. Of course this also makes for attention-getting headlines.

We know this to be the case for television weather-reporting, where summer temperatures are made to appear warmer through use of a humidity index, and winter temperatures made colder by using a wind-chill index.

It turns out that Canadians do something similar for certain earthquakes by using a Nuttli magnitude index. Apparently, "Canadian seismologists will often refer to the Richter magnitude whereas strictly speaking the seisms that occur in Eastern Canada are measured according to the Nuttli magnitude. An exception exists for the very small earthquakes of the Charlevoix Region, where the Richter scale is used." An exception with an exception. Nice.

We know this to be the case for television weather-reporting, where summer temperatures are made to appear warmer through use of a humidity index, and winter temperatures made colder by using a wind-chill index.

It turns out that Canadians do something similar for certain earthquakes by using a Nuttli magnitude index. Apparently, "Canadian seismologists will often refer to the Richter magnitude whereas strictly speaking the seisms that occur in Eastern Canada are measured according to the Nuttli magnitude. An exception exists for the very small earthquakes of the Charlevoix Region, where the Richter scale is used." An exception with an exception. Nice.

## Sunday, April 21

### Emmo W.

Emmo W. was the nom de plume (by way of M.O.W.) of a Melvin Oscar Wellman. Melvin was born 18 January 1881 in the township of Danby, Michigan (roughly west-northwest of Lansing). In 1910 we find him in Charlotte with a wife and two sons; and from 1920 on, in Lansing.

I am indebted to Melvin's grandson, William W. Wellman, for providing me with additional information. He writes:

*I spent a lot of time fishing with my grandfather, into my mid-teens.*

*Melvin was an avid fisherman who made split bamboo fly rods, for himself and both of his sons. Every summer in the early 1900s, he took his family by rail to Petoskey, Michigan to spend weekends fishing local small lakes accessible by train. During the week, he barbered in a popular barber shop, McCarthy's, where he may have cut Ernest Hemmingway's hair.*

*Melvin was the inventor of several camping products but never applied for patents. He used an early hearing aid and founded the Michigan Better Hearing Association, now known as the Michigan Speech-Language-Hearing Association.*

*Even though my grandfather only had an 8th grade formal education, he was the smartest person I ever knew.*

*He was an avid reader of English, history, and puzzle books.*

Melvin was also a regular contributor in the 1940s to

*The Enigma*(a publication of the U.S. National Puzzlers' League) and is credited with introducing therein, in March 1945, the spoonergram. In the April 1948 issue, he gave us

*this*enigma:

And here is how the mysterious Dr. Matrix (Martin Gardner narrating) paraphrased it in Scientific American in January 1960 (page 154):

*"11 plus 2 minus 1 is 12. Let me show you how this works out with letters." He moved to the blackboard and chalked on it the word ELEVEN. He added TWO to make ELEVEN-TWO, then he erased the letters of ONE, leaving ELEVTW. "Rearrange those six letters," he said, "and they spell TWELVE."*

The anagram ELEVEN + TWO = TWELVE + ONE is well known in word-play circles, though generally stated without attribution. Now you know from whence it came.

Here is a photo of Melvin and his wife Lucy, later in life. Melvin died 7 October 1955.

## Friday, April 19

### Manhunt marathon

When (this evening) I finally sat down to

*watch*television (instead of just listening to it from my computer room), I augmented CNN with a Google+ feed of #Watertown on my iPad. When someone posted that the suspect was in a*boat*, I took it for a troll (a little contextual information would have helped) — until CNN reported it as well, some minutes later. News of the capture, likewise, preceded CNN's reporting of it by four or five minutes. Of course it is difficult to ascertain which posts offer*credible*information but as long as one maintains one's usual sense of skepticism, a several minutes advantage in an unfolding news event is manhunt manna.## Tuesday, April 16

### How far apart were the two Boston marathon bombing sites?

"50 to 100 yards" according to Boston Police Commissioner Ed Davis in a news conference. A lot of newspapers printed this as though it might be true. Canadian media settled on 100 meters as a good-enough approximation. I was pleasantly surprised that Wikipedia (when I checked earlier today) had the blasts occurring "within 550 feet" of each other — somewhat closer to the truth.

The blast locations are no secret: There are plenty of photographs. The first happened in front of Lens Crafters at 699 Boylston; the second, in front of Forum at 755 Boylston. Some folk tried to place the first blast in front of Marathon Sports, next-door to Lens Crafters, but the damage done to the Lens Crafters facade speaks for itself.

So we know each location within a meter or two. Using Google street view to familiarize oneself with the street and building appearances, one can — in Google Earth — situate correctly both locations using the ruler tool: 183 meters, give or take.

The blast locations are no secret: There are plenty of photographs. The first happened in front of Lens Crafters at 699 Boylston; the second, in front of Forum at 755 Boylston. Some folk tried to place the first blast in front of Marathon Sports, next-door to Lens Crafters, but the damage done to the Lens Crafters facade speaks for itself.

So we know each location within a meter or two. Using Google street view to familiarize oneself with the street and building appearances, one can — in Google Earth — situate correctly both locations using the ruler tool: 183 meters, give or take.

## Wednesday, April 10

### Composition

*Primes, primes, every where,*

*Was all the bard did think;*

*Primes, primes, every where,*

*But nary one in link.**

This base-ten sequence exhibits an absence of prime

*linked primes*(that is, the concatenation of

*any number of consecutive terms*) in an infinite sea of primes:

2, 5, 11, 13, 29, 31, 17, 19, 43, 7, 37, 41, 71, 47, 67, 89, 3, 101, 23, 109, 59, 83, 103, 73, 107, 157, 53, 127, 149, 61, 131, 139, 79, 163, 191, 193, 97, 113, 137, 167, 211, 181, ...

Such sequences are not rare, this one being the lexicographically first. Here is the base-two analogue:

2, 5, 17, 13, 11, 23, 3, 19, 7, 53, 37, 31, 47, 29, 43, 59, 41, 73, 67, 83, 89, 61, 79, 71, 107, 97, 127, 131, 101, 113, 151, 103, 137, 109, 167, 179, 139, 227, 149, 191, 157, 193, ...

Here is one that works in either base-two

*or*base-ten:

2, 5, 17, 43, 7, 23, 19, 127, 11, 41, 157, 101, 13, 131, 3, 211, 37, 149, 163, 173, 31, 107, 229, 29, 89, 67, 109, 223, 73, 193, 47, 79, 59, 71, 179, 191, 151, 97, 269, 139, 277, 227, ...

And this one works in any base from two to ten:

2, 229, 131, 263, 37, 421, 491, 223, 911, 127, 167, 383, 1187, 401, 31, 15307, 701, 971, 2797, 3, 8741, 571, 5477, 6037, 619, 859, 6359, 353, 2659, 311, 3851, 379, 7193, 7993, 3319, 653, 691, 13441, 661, 1579, 7541, 1987, ...

*

*Primes of the Ancient Mariner*

## Monday, March 25

### Hoffman's packing puzzle

This is a photo of my Bill Cutler version of Hoffman's packing puzzle which I have had socked away in a cabinet for the better part of thirty-five years. The blocks are 15x18x22 deci-inches and the frame, 55 cubed. Bill's pieces (each one composed of a 7.5x18x22 doublet) sport some rough saw-cut ends and sides. The frame was originally a box, but one of its sides warped and I decided that it would look better with

Dean Hoffman thought up the packing problem in 1978 (Bill Cutler thinks it was 1976: see #8 here) and wrote about it in David Klarner's The Mathematical Gardner (1981, pages 212-225). Elwyn Berlekamp, John Conway, and Richard Guy covered it in their Winning Ways (1982, volume 2: pages 739-740, 804-806; 2004 second edition, volume 4: pages 847-848, 913-915). Alexey Spiridonov published a very nice article about it and its solutions in 2003 and posited an approach to solving the four-dimensional analogue. I don't know if this has yet been accomplished. George Miller, on one of his Puzzle Palace pages, has Donald Knuth searching (in 2004) for solutions to a 3x4x5 version of Hoffman's problem and finding three where one could squeeze an extra

*that*and another one of its sides removed. Fine woodworking versions have been created by Trevor Wood and John Devost. Gemani Games and Puzzles sells a version in Samanea.Dean Hoffman thought up the packing problem in 1978 (Bill Cutler thinks it was 1976: see #8 here) and wrote about it in David Klarner's The Mathematical Gardner (1981, pages 212-225). Elwyn Berlekamp, John Conway, and Richard Guy covered it in their Winning Ways (1982, volume 2: pages 739-740, 804-806; 2004 second edition, volume 4: pages 847-848, 913-915). Alexey Spiridonov published a very nice article about it and its solutions in 2003 and posited an approach to solving the four-dimensional analogue. I don't know if this has yet been accomplished. George Miller, on one of his Puzzle Palace pages, has Donald Knuth searching (in 2004) for solutions to a 3x4x5 version of Hoffman's problem and finding three where one could squeeze an extra

*28th*block into a 12-cubed frame!## Sunday, March 3

### Coming out

The stove is in, Hazel's pen has been rebooted, and she has been let out to scamper around the new floor and get reacquainted with the furniture. Sharing a kitchen with a rabbit might not be everyone's cup of tea but it is manageable and allows us to spend more time with her than we might otherwise.

Also, the kitchen opens onto our back deck and I have been known to let Hazel out occasionally to enjoy the great outdoors. Catherine is

Also, the kitchen opens onto our back deck and I have been known to let Hazel out occasionally to enjoy the great outdoors. Catherine is

*disinclined*to do so, put off by the incessant self-scratching behavior of our rodent visitors.## Saturday, March 2

### First light

Our kitchen makeover involved painting the walls, laying down a new floor, and installing a new stove. Here is Catherine lighting it. It still needs to be pushed back into its space between the cabinets but we will wait for help so as not to scratch the floor.

The oven part of our previous stove had not been used in a very long time because, many years ago, mice had gotten into it and made it their abode. Catherine has been using a mini-oven to do her baking since but

The oven part of our previous stove had not been used in a very long time because, many years ago, mice had gotten into it and made it their abode. Catherine has been using a mini-oven to do her baking since but

*I*have not done any baking at all, which is a shame because I used to make some decent cakes. I will have to bake one for Catherine's upcoming birthday!## Sunday, February 24

### A mathematically crippling deformity

I've written about Stewart Coffin's Convolution puzzle before. John Rausch has a partially assembled Convolution on his Puzzle World website, to which I have added x,y,z axes and red line that goes from (2,2,2) to (2,2,3):

Three pieces (F, D, and G) need to be added to the assemblage. The black

*cubies*end up in the corners of the finished cube. Also, F and G are seriously foreshortened (what look like one-cubie extensions going to the back are actually

*two-cubie*extensions). The number of cubies in each of the three unassembled pieces is 9 (to fit into the 27 empty spaces of the unfinished cube). Can you complete the construction in your head?

I'm going to describe in some detail the 'rotation' that makes Convolution such an interesting puzzle. Bill Cutler once told me that he had

*done*the math, but any mention of the rotation today (as far as I know) neglects to provide any details beyond pointing out that one or more cubie edges are slightly rounded to allow it.

It is D that is the next piece to get added to the unfinished assemblage. Because there is some degree of freedom in how initially this happens, I will describe instead the reverse process: how D is removed from an assembled cube where G and F (in that order) have already been removed. In its final resting place, D's black cubie will sit at the (0,4,4) corner of the finished cube. The other end of D (the final cubie of the three-cubie straight arm) will sit underneath the black cubie whose corner is at (4,0,4). How it got there is of course what this blog entry is all about.

All the action happens in a space parallel to the xy plane (from z=2 to z=4). To dislodge D, first it is pushed down the y-axis by one unit. Having done so, the view of D from above the xy plane will be:

I've added a grid at z=4 showing a few coordinates and two points: A at (2,1) and B at (3,2). Now, the 'rotation': What happens is that A slides along y=1 toward the right while, at the same time, B slides along x=3 toward the top. How long is the slide? Somewhere between D's x=3 line reaching the point (4,1) and D's (4,2) corner reaching the line y=3. Within this range, piece D may be removed by lifting it up the z-axis.

Is this the only edge that is compromised? Convolution designer Stewart Coffin said that the assembly "requires a rotation, which is not possible unless certain edges are rounded ever so slightly". [The Puzzling World of Polyhedral Dissections, 1990, page 52; or here]. So he implies

*more*than one. But this may have been a woodworking — not a mathematical — depiction. Once the (red) edge has been rounded, it allows the rotation/slide to proceed slightly

*before*the one-unit push down the y-axis is fully complete,

*but*at the expense of the lower-level B-point edge on the D piece.

In that 1990 reference, Coffin asks: "Can any reader devise a way to correct this mechanically slight but mathematically crippling deformity in an otherwise satisfactory design?" I discovered this week that such an improvement exists.

## Monday, January 14

### Eight- and nine-letter words in base-26 pi: II

The 32 eight- and nine-letter words from Peter Norvig's Google word-count file that were found in half a billion letters of the base-26 π-code representation of the real digits of π have now been supplemented by finds from larger data sets (not restricted by Norvig's at-least-100000-mentions cutoff criterion).

Specifically, my search resulted in an

3095146 Armagnac

5204508 reformist

5446573 fabledom

12767754 pediatry

23893131 keratoma

26460749 plastics

30620629 Batavian

34355657 sailorly

38729316 hatbrush

46803099 Gemmingia

49292523 raisonné

52221111 beakerman

52374041 infandous

62288036 Altamont

68386037 handsome

77174448 piquance

80344659 spraints

85983887 ticktock

95489940 freewill

104799581 glassful

119398927 obligate

122636295 derriere

144023162 tarragon

145410250 Pannonic

148864411 aphicide

160285943 conveyer

168667826 hockshin

179537813 caraboid

186970055 lineages

194941942 symbolic

203750087 drawling

204682494 subreguli

213927339 aquiform

220130527 pajamaed

223387624 blurbist

227698058 Gederite

232625291 moromancy

233706360 Brockway

238312955 homicide

241593178 aularian

244832756 coenzyme

245790734 clinamen

248977229 offenses

253217633 somewise

258077020 masslike

265316858 draftily

270498733 puncheon

290930240 friction

291953969 Judentum

296560665 torpidity

298503676 eddyroot

308820127 engaging

309864510 octapody

310692296 Alabaman

317941229 outgrown

324802306 dartlike

326873656 hayfield

327954809 jamboree

330311394 grubbily

331195875 monodont

334661344 venially

339119974 panderly

341079873 magneton

358147952 benzamide

362326813 autopsic

378333440 bookings

379470966 assenter

400726498 cardanic

414326761 immotive

426642188 slubbery

428186515 noblesse

433412589 inertial

440674037 ephebeum

442091394 unkilled

443277601 bioplasm

444201817 Crataeva

452027527 driftlet

454659011 pineland

460082749 loathness

467631243 prickish

468685858 pyroboric

475910828 Mersenne

476984745 stigmatic

479595795 Vallarta

480168788 sunblink

483460192 atmiatry

487934346 copyists

488079020 Assyrian

499784890 southron

Specifically, my search resulted in an

*additional*55 eight-letter and 9 nine-letter hits. Of the eight-letter finds, I decided to reject**goyetian**,**rosaruby**,**avanious**,**fleyland**,**commoney**,**scambler**, and**tortuose**. I immediately recognized the nine-letter**beakerman**as a word I had come across in December 2003 (at the time, I had saved a picture of muppet*Beaker*and had given it that name) but I have no corresponding Mathematica notebook to document the find and have only a vague recollection of extending my year-2000 calculation. At any rate, I had struggled back then with recognizing*beakerman*as a legitimate word and I did so again now. (I have kept it.) So, 32+55-7+9 = 89 words:3095146 Armagnac

5204508 reformist

5446573 fabledom

12767754 pediatry

23893131 keratoma

26460749 plastics

30620629 Batavian

34355657 sailorly

38729316 hatbrush

46803099 Gemmingia

49292523 raisonné

52221111 beakerman

52374041 infandous

62288036 Altamont

68386037 handsome

77174448 piquance

80344659 spraints

85983887 ticktock

95489940 freewill

104799581 glassful

119398927 obligate

122636295 derriere

144023162 tarragon

145410250 Pannonic

148864411 aphicide

160285943 conveyer

168667826 hockshin

179537813 caraboid

186970055 lineages

194941942 symbolic

203750087 drawling

204682494 subreguli

213927339 aquiform

220130527 pajamaed

223387624 blurbist

227698058 Gederite

232625291 moromancy

233706360 Brockway

238312955 homicide

241593178 aularian

244832756 coenzyme

245790734 clinamen

248977229 offenses

253217633 somewise

258077020 masslike

265316858 draftily

270498733 puncheon

290930240 friction

291953969 Judentum

296560665 torpidity

298503676 eddyroot

308820127 engaging

309864510 octapody

310692296 Alabaman

317941229 outgrown

324802306 dartlike

326873656 hayfield

327954809 jamboree

330311394 grubbily

331195875 monodont

334661344 venially

339119974 panderly

341079873 magneton

358147952 benzamide

362326813 autopsic

378333440 bookings

379470966 assenter

400726498 cardanic

414326761 immotive

426642188 slubbery

428186515 noblesse

433412589 inertial

440674037 ephebeum

442091394 unkilled

443277601 bioplasm

444201817 Crataeva

452027527 driftlet

454659011 pineland

460082749 loathness

467631243 prickish

468685858 pyroboric

475910828 Mersenne

476984745 stigmatic

479595795 Vallarta

480168788 sunblink

483460192 atmiatry

487934346 copyists

488079020 Assyrian

499784890 southron

A

At index 115577805 is the string

*ten-letter*word is not found in this range — unless we are willing to allow*back*words:
..rlivetumsnwlieeq

**remonobons**eetacejbfepewqxd..At index 115577805 is the string

**remonobons**, which is**snobonomer**in reverse. William Makepeace Thackeray used this word in his satirical writing: "Some telescopic philosopher will arise one day, some great Snobonomer, to find the laws of the great science which we are now merely playing with, and to define, and settle, and classify that which is at present but vague theory, and loose, though elegant assertion."## Saturday, January 12

### Eight- and nine-letter words in base-26 pi

In my previous post, I provided some English number words that appear in Mike Keith's base-26 π-code representation of the real digits of π. This entry is about other words.

In 2000, I found the eight-letter

A search resulted in 35 eight-letter and 2 nine-letter hits. I dismissed

In the following, I have capitalized the words (including a German one) that I felt needed capitalization and added an accent on one of the three French words.

3095146 Armagnac

5204508 reformist

26460749 plastics

30620629 Batavian

49292523 raisonné

62288036 Altamont

68386037 handsome

95489940 freewill

119398927 obligate

122636295 derriere

144023162 tarragon

160285943 conveyer

186970055 lineages

194941942 symbolic

203750087 drawling

233706360 Brockway

238312955 homicide

244832756 coenzyme

248977229 offenses

290930240 friction

291953969 Judentum

308820127 engaging

317941229 outgrown

327954809 jamboree

378333440 bookings

428186515 noblesse

433412589 inertial

475910828 Mersenne

476984745 stigmatic

479595795 Vallarta

487934346 copyists

488079020 Assyrian

32 words: It's a start.

In 2000, I found the eight-letter

**armagnac**at index 3095146. Now, with half a billion strung-together letters (one hundred times the "real estate") at my disposal, I expected to find many more eight-letter words and, hopefully, some larger ones as well. I used eight- and nine-letter words culled from Peter Norvig's Google word-count file made available in his recent English Letter Frequency Counts essay.A search resulted in 35 eight-letter and 2 nine-letter hits. I dismissed

**gruening**,**schreber**,**brentano**, and**hillquit**for being surnames only. (I kept**mersenne**because of its adjectival usefulness in mathematics.) I also excluded**thoufand**— an alternate, incorrect version of*thousand*resulting from the difficulty of distinguishing a*long*s from an f in old-English typography. That*thoufand*had 158819 mentions in Norvig's data set amply demonstrates his list's limitations (and questions his conclusions).In the following, I have capitalized the words (including a German one) that I felt needed capitalization and added an accent on one of the three French words.

3095146 Armagnac

5204508 reformist

26460749 plastics

30620629 Batavian

49292523 raisonné

62288036 Altamont

68386037 handsome

95489940 freewill

119398927 obligate

122636295 derriere

144023162 tarragon

160285943 conveyer

186970055 lineages

194941942 symbolic

203750087 drawling

233706360 Brockway

238312955 homicide

244832756 coenzyme

248977229 offenses

290930240 friction

291953969 Judentum

308820127 engaging

317941229 outgrown

327954809 jamboree

378333440 bookings

428186515 noblesse

433412589 inertial

475910828 Mersenne

476984745 stigmatic

479595795 Vallarta

487934346 copyists

488079020 Assyrian

32 words: It's a start.

## Wednesday, January 9

### A googol in pi

In 1999, Mike Keith worked on a base-26 representation of the number π, wherein the digit

π = d.drsqlolyrtrodnlhnqtgkudqgtuirxneqbckbszivqqvgdmelmuexroiqiyalvuz..

First, I will set up indexing terminology so that there is no misunderstanding about exactly

Mike and I looked for

**0**is replaced with the letter**a**, the digit**1**with the letter**b**, .., the digit**25**with the letter**z**:π = d.drsqlolyrtrodnlhnqtgkudqgtuirxneqbckbszivqqvgdmelmuexroiqiyalvuz..

First, I will set up indexing terminology so that there is no misunderstanding about exactly

*where*things are. I am partial (*dictatorial*, because of my work with continued fractions) to dropping, here, the whole number**d**and indexing the subsequent,*fractional***drs..**{1,2,3,..}. Thus the expression**lol**is (*begins*) at index 5.Mike and I looked for

*large*words in the base-26 π expansion and*number*words of all sizes. In 2000, it took me 127 hours to calculate 5 million terms. My present setup accomplishes it in 13 seconds! So it didn't take me long to*up*the ante and I can now report that I have found a**googol**at index 454315613. Already appeared by then are the English number words for 1, 4, 2, 6, 10, 5, 9, 0, 7, 3, 50, 40, 8, 60, 11, 90, 80, 12, 30 (*first appearances*, respectively at indices 10087, 11324, 13463, 14295, 15276, 64838, 175372, 389247, 786244, 1556763, 2300987, 8879098, 9202330, 9946442, 33027856, 126003234, 126348794, 238426469, 389952198). Here are all twenty first appearances, as one would see them in situ:
..datnmvbwezlfygfl

**one**bugwcjooatbjbskh..
..wzlwdhgbhphxeqsq

**four**djjmmikzqwvivabm..
..pceoeiigwyqtmaus

**two**amxyatxhqxoyzbno..
..ifnuetcfdxrwipod

**six**yavsnasweexagllo..
..ectasqgcghjmzirr

**ten**fjiphdkdoumqzpjt..
..dwbfgegkqjwkubjv

**five**wgjlbihzmwgalifm..
..jyhdeevmsxmbxtbi

**nine**suzrnrzsxosctuxm..
..efzoqitnhuljsgzt

**zero**vlptejdfwretaara..
..hwdvoynebabsyjkd

**seven**aabgjthpqgapqnrg..
..cgakthjgxsmoldok

**three**apdiuhncqcjgsmnt..
..gfaoforthfmdhgkz

**fifty**xiboudqllqzqlcmh..
..obptjczwqdirtnvm

**forty**igfxwmybvztjkhbc..
..hubksmltneixkvph

**eight**mgsxrwaqokrdcpnw..
..ttsybfashnuibvli

**sixty**tdchrjcvvzazzivu..
..wltescttnomszton

**eleven**fyifyclxdcacsfta..
..qpddtdyuitfganmg

**ninety**hecteviwajvcosrv..
..kvvtnvsdypwsumzv

**eighty**sstqdeajfbyxrrde..
..iuwxlmueqjjzpbcm

**twelve**efcsszipcnmifmzp..
..ssoshdnbsnembdkg

**thirty**qwbfxohiadwgtvya..
..dnihymgxncwxfhsd

**googol**xvodlfjhbzlnrucr..
The word **google**does

*not*appear in the first half billion terms.## Tuesday, January 8

### Letter frequencies

*The Great Gatsby*:

**E**TAONISHRDLUMWGCYFBPKVJXQZ

The great

*Gadsby*: AOINTSLHRUDGYWCFBMPKVJZXQ

**E**

## Monday, January 7

### The principle of Laplace

Marcello Truzzi's "an extraordinary claim requires extraordinary proof" is generally credited to Pierre-Simon Laplace (via Carl Sagan) in the form of "the weight of evidence for an extraordinary claim must be proportioned to its strangeness". I did a little Google searching in an attempt to source this and (finally) came up with William McDougall's

"We are so far from knowledge of all natural agencies and of their diverse modes of action, that it would not be philosophic to deny any phenomena simply because they are inexplicable in the present state of our knowledge. But we ought to examine them with an attention the more painstaking, the more difficult it may seem to accept them as real."

McDougall says the quote was made "by the great exponent of the mechanical view of the universe, Laplace" and that a Théodore Flournoy would call it "the principle of Laplace" and state it briefly as "the weight of the evidence should be proportional to the strangeness of the alleged facts". Ironically then, it would appear that the quote generally attributed to Laplace is actually a quote of Flournoy, a one-time believer of telekinesis, telepathy, and clairvoyance!

*Outline of Abnormal Psychology*(1926), which has (on page 508):"We are so far from knowledge of all natural agencies and of their diverse modes of action, that it would not be philosophic to deny any phenomena simply because they are inexplicable in the present state of our knowledge. But we ought to examine them with an attention the more painstaking, the more difficult it may seem to accept them as real."

McDougall says the quote was made "by the great exponent of the mechanical view of the universe, Laplace" and that a Théodore Flournoy would call it "the principle of Laplace" and state it briefly as "the weight of the evidence should be proportional to the strangeness of the alleged facts". Ironically then, it would appear that the quote generally attributed to Laplace is actually a quote of Flournoy, a one-time believer of telekinesis, telepathy, and clairvoyance!

## Friday, January 4

### Mate the royal couple

This was the title of Valentin Albillo's 1997 unsolved position #91. In 2003, Vincent Lejeune had ChessMaster 9000 declare e4 as a mate-in-11. I wanted to see what a modern, off-the-shelf chess engine would do with this position.

**3qk3/8/8/8/8/8/PPPPPPPP/RNBQKBNR w**

The program I used was HIARCS Chess Explorer. The analysis engine is Deep HIARCS 14 WCSC. Because I am running several number-crunching applications concurrently, I did not expect to see full performance on my late-2012 iMac. Nevertheless, in its default hash-table-size setting, it managed a mate-in-11 for e4 in a few hours.

I restarted the 32-bit chess engine with an increased hash-table-size of 2 GB (the maximum possible). Unexpectedly, it did not discover the e4 mate as quickly, but (as compensation, perhaps) noted that Nc3 was mate-in-11 as well. I have kept the evaluation running to see what the other eighteen lines might produce. Eventually, they are mate-in-12 for twelve (d4, e3, h4, a4, b3, c3, Nh3, Nf3, h3, d3, b4, c4) of them, mate-in-13 for five (a3, Na3, g4, g3, f4), and mate-in-14 for f3. HIARCS is not a mate solver, so these numbers should be seen as mate-in-

*at-most*with a slight improvement possible, though not entirely expected.## Tuesday, January 1

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