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A B R A C

A B R A

A B R

A B

A
On pp. 39-40 he describes and illustrates an inscription on the Stele of Moschion from Egypt, c300. This is a 39 x 39 square with a Greek text from the middle to the corner, e.g. like the example in the following entry. The text reads: ΟΥIΡIΔIΜΟΥΧIΩΝΥΓIΑΥΘΕIΥΤΟΝΠΟΔΑIΑΤΠΕIΑIΥ which means: Moschion to Osiris, for the treatment which cured his foot. Millington does not ask for the number of ways to read the inscription, which is 4 BC(38,19) = 14 13810 55200.

Curiosities for the Ingenious selected from The most authentic Treasures of E D C D E

Nature, Science and Art, Biography, History, and General Literature. D C B C D

(1821); 2nd ed., Thomas Boys, London, 1822. Remarkable epitaph, C B A B C

p. 97. Word diamond extended to a square, based on 'Silo Princeps Fecit', D C B C D

with the ts at the corners. An example based on 'ABCDEF' is shown E D C D E

at the right. Says this occurs on the tomb of a prince named Silo at the

entrance of the church of San Salvador in Oviedo, Spain. Says the epitaph can be read in 270 ways. I find there are 4 BC(16, 8) = 51490 ways.

In the churchyard of St. Mary's, Monmouth, is the gravestone of John Rennie, died 31 May 1832, aged 33 years. This has the inscription shown below. Further down the stone it gives his son's name as James Rennie. Apparently an N has been dropped to get a message with an odd number of letters. I have good photos. Nothing asks for the number of ways of reading the inscription. I get 4 BC(16,9) = 45760 ways.

eineRnhoJsJohnRenie

ineRnhoJsesJohnReni

neRnhoJseiesJohnRen

eRnhoJseiliesJohnRe

RnhoJseileliesJohnR

nhoJseilereliesJohn

hoJseilerereliesJoh

oJseilereHereliesJo

hoJseilerereliesJoh

nhoJseilereliesJohn

RnhoJseileliesJohnR

eRnhoJseiliesJohnRe

neRnhoJseiesJohnRen

ineRnhoJsesJohnReni

eineRnhoJsJohnRenie
Nuts to Crack I (1832), no. 200. The example from Curiosities for the Ingenious with 'SiloPrincepsFecit', but no indication of what is wanted -- perhaps it is just an amusing picture.

W. Staniforth. Letter. Knowledge 16 (Apr 1893) 74-75. Considers 1 2 3 4 5 6

"figure squares" as at the right. "In how many different ways may 2 3 4 5 6 7

the figures in the square be read from 1 to 11 consecutively?" 3 4 5 6 7 8

He computes the answers for the n x n case for the first few 4 5 6 7 8 9

cases and finds a recurrence. "Has such a series of numbers any 5 6 7 8 9 10

mathematical designation?" The editor notes that he doesn't 6 7 8 9 10 11

know.

J. J. Alexander. Letter. Knowledge 16 (May 1893) 89. Says Staniforth's numbers are the sums of the squares of the binomial coefficients BC(n, k), the formula for which is BC(2n, n). Editor say he has received more than one note pointing this out and cites a paper on such figure squares by T. B. Sprague in the Transactions of the Royal Society of Edinburgh -- ??NYS, no more details provided.

Loyd. Problem 12: The temperance puzzle. Tit Bits 31 (2 & 23 Jan 1897) 251 & 307. Red rum & murder. = Cyclopedia, 1914, The little brown jug, pp. 122 & 355. c= MPSL2, no. 61, pp. 44 & 141. Word diamond based on 'red rum & murder', i.e. the central line is redrum&murder. He allows a diagonal move from an E back to an inner R and this gives 372 paths from centre to edge, making 372² = 138,384 in total.

Dudeney. Problem 57: The commercial traveller's puzzle. Tit Bits 33 (30 Oct & 20 Nov 1897) 82 & 140. Number of routes down and right on a 10 x 12 board. Gives a general solution for any board.

Dudeney. A batch of puzzles. The Royal Magazine 1:3 (Jan 1899) 269-274 & 1:4 (Feb 1899) 368-372. A "Reviver" puzzle. Complicated pattern based on 'reviver'. 544 solutions.

Dudeney. Puzzling times at Solvamhall Castle. London Magazine 7 (No. 42) (Jan 1902) 580 584 & 8 (No. 43) (Feb 1902) 53-56. The amulet. 'Abracadabra' in a triangle with A at top, two B's below, three R's below that, etc. Answer: 1024. = CP, 1907, No. 38, pp. 64-65 & 190. CF Millington at beginning of this section.

Dudeney. CP. 1907.

Prob. 30: The puzzle of the canon's yeoman, pp. 55-56 & 181-182. Word diamond based on 'was it a rat I saw'. Answer is 63504 ways. Solution observes that for a diamond of side n+1, with no diagonal moves, the number of routes from the centre to an edge is 4(2ⁿ-1) and the number of ways to spell the phrase is this number squared. Analyses four types with the following central lines: A   'yoboy'; B - 'level'; C - 'noonoon'; D   'levelevel'.

In A, one wants to spell 'boy', so there are 4(2ⁿ-1) solutions.

In  B, one wants to spell 'level' and there are [4(2ⁿ-1)]² solutions.

In C, one wants to spell 'noon' and there are 8(2ⁿ-1) solutions.

In D, one wants to spell 'level' and there are complications as one can start and finish at the edge. He obtains a general formula for the number of ways. Cf Loyd, 1914.

Prob. 38: The amulet, pp. 64-65 & 190. See: Dudeney, 1902.

Pearson. 1907. Part II: A magic cocoon, p. 147. Word diamond based on 'cocoon', so the central line is noocococoon. Because one can start at the non central Cs, and can go in as well as out, I get 948 paths. He says 756.

Loyd. Cyclopedia. 1914. Alice in Wonderland, pp. 164 & 360. = MPSL1, no. 109, pp. 107 & 161 162. Word diamond based on 'was it a cat I saw'. Cf Dudeney, 1907.

Dudeney. AM. 1917.

Prob. 256: The diamond puzzle, pp. 74 & 202. Word diamond based on 'dnomaidiamond'. This is type A of his discussion in CP and he states the general formula. 252 solutions.

Prob. 257: The deified puzzle, pp. 74-75 & 202. Word diamond based on 'deifiedeified'. This is type D in CP and has 1992 solutions. He says 'madamadam' gives 400 and 'nunun' gives 64, while 'noonoon' gives 56.

Prob. 258: The voter's puzzle, pp. 75 & 202. Word diamond built on 'rise to vote sir'. Cites CP, no. 30, for the result, 63504, and the general formula.

Prob. 259: Hannah's puzzle, pp. 75 & 202. 6 x 6 word square based on 'Hannah' with Hs on the outside, As adjacent to the Hs and four Ns in the middle. Diagonal moves allowed. 3468 ways.

Wood. Oddities. 1927. Prob. 44: The amulet problem, p. 39. Like the original ABRACADABRA triangle, but with the letters in reverse order.

Collins. Book of Puzzles. 1927. The magic cocoon puzzle, pp. 169-170. As in Pearson.

Loyd Jr. SLAHP. 1928. A strolling pedagogue, pp. 38 & 97. Number of routes to opposite corner of a 5 x 5 array of points.

D. F. Lawden. On the solution of linear difference equations. MG 36 (No. 317) (Sep 1952) 193-196. Develops use of integral transforms and applies it to find that the number of king's paths going down or right or down right from (0, 0) to (n, n) is P_n(3) where P_n(x) is the Legendre polynomial.

Leo Moser. King paths on a chessboard. MG 39 (No. 327) (Feb 1955) 54. Cites Lawden and gives a simpler proof of his result P_n(3).

Anon. Puzzle Page: Check this. MTg 36 (1964) 61 & 27 (1964) 65. Find the number of king's routes from corner to corner when he can only move right, down or right down. Gets 48,639 routes on 8 x 8 board.

Ripley's Puzzles and Games. 1966. P. 32. Word diamond laid out differently so A A A

one has to read from one side to the opposite side. Rotating by 45^o, one gets B B 

the pattern at the right for edge three. One wants the number of ways to C C C

read ABCDEF. In general, when the first line of As has n positions, D D 

the total number of ways to reach the first row is n. For each successive E E E

row, the total number is alternately twice the number for the previous row less F F 

twice the end term of that row or just twice the the number for the previous

row. In our example with n = 3, the number of ways to reach the second row is 4 = 2x3 - 2x1. The number of ways to reach the third row is 8 = 2x4. The number of ways to reach the fourth row is 12 = 2x8 - 2x2, then we get 24 = 2 x 12; 36 = 2x24   2x6. It happens that the first n end terms are the central binomial coefficients BC(2k,k), so this is easy to calculate. I find the total number of routes, for n = 2, 3, ..., 7, is 4, 18, 232, 1300, 6744, 33320, the last being the desired and given answer for the given problem.

Pál Révész. Op. cit. in 5.I.1. 1969. On p. 27, he gives the number of routes for a king moving forward on a chessboard and a man moving forward on a draughtsboard.

Putnam. Puzzle Fun. 1978. No. 8: Level - level, pp. 3 & 26. Form a wheel of 16 points labelled LEVELEVELEVELEVE. Place 4 Es inside, joined to two consecutive Vs and the intervening L. Then place a V in the middle, joined to these four Es. How many ways to spell LEVEL? He gets 80, which seems right.