Sources page biographical material

M. Adams. Puzzles That Everyone Can Do. 1931. o o

Prob. 37, pp. 22 & 134: 20 counter problem. Given the pattern of o o

20 counters at the tight, 'how many perfect squares are o o o o o o

contained in the figure.' This means having their vertices o o o o o o

at counters. There are surprisingly more than I expected. o o

Taking the basic spacing as one, one can have squares of o o

edge 1, 2, 5, 8, 13, giving 21 squares in all.

He then asks how many counters need to be removed in order to destroy all the squares? He gives a solution deleting six counters.

Prob. 217, pp. 83 & 162: Match squares. He gives 10 matches making a row of three equal squares and asks you to add 14 matches to form 14 squares. The answer is to make a 3 x 3 array of squares and count all of the squares in it.

J. C. Cannell. Modern Conjuring for Amateurs. C. Arthur Pearson, London, nd [1930s?]. Counting the squares, pp. 84-85. As in Blyth.

Indoor Tricks and Games. Success Publishing, London, nd [1930s??]. Square puzzle, p. 62. Start with a square and draw its diagonals and midlines. Join the midpoints of the sides to form a second level square inscribed in the first level original square. Repeat this until the 9th level. How many squares are there? Given answer is 16, but in my copy someone has crossed this out and written 45, which seems correct to me.

Meyer. Big Fun Book. 1940. No. 9, pp. 162 & 752. Draw four equidistant horizontal lines and then four equidistant verticals. How many squares are formed? This gives a 3 x 3 array of squares, but he counts all sizes of squares, getting 9 + 4 + 1 = 14. (Also in 7.AU.)

Foulsham's New Party Book. Foulsham, London, nd [1950s?]. P. 103: How many squares? 4 x 4 board with some extra diagonals giving one extra square.

Although there are few references before this point, the puzzle idea was pretty well known and occurs regularly in the children's puzzle books of Norman Pulsford which start c1965. He gives various irregular patterns and asks for the number of triangles or squares.

Jonathan Always. Puzzles to Puzzle You. Op. cit. in 5.K.2. 1965. No. 140: A surprising answer, pp. 43 & 90. 4 x 4 chessboard with four corner cells deleted. How many rectangles are there?

Anon. Puzzle page: Strictly for squares. MTg 30 (1965) 48 & 31 (1965) 39 & 32 (1965) 39. How many squares on a chessboard? First solution gets S(8) = 1 + 4 + 9 + ... + 64 = 204. Second solution observes that there are skew squares if one thinks of the board as a lattice of points and this gives S(1) + S(2) + ... + S(8) = 540 squares.

G. A. Briggs. Puzzle and Humour Book. Published by the author, Ilkley, 1966.

Prob. 2/11, pp. 22 & 74. 4 x 4 array of squares bordered on two sides by bricks 1 x 2, 1 x 3, 2 x 1, 2 x 1. Count the squares and the rectangles. Gets 35 and 90.

Prob. 2/14, pp. 23 & 75. Pattern of squares making the shape of a person -- how many squares in it?

Ripley's Puzzles and Games. 1966. Pp. 72-73 have several problems of counting squares.

Item 4. Consider a 3 x 3 array of squares with their diagonals drawn. The solution says this has 30 squares. I get 31, but perhaps they weren't counting the whole figure. I have computed the total number of squares for an n x n array and get (2n³ + n²)/2 squares for n even and (2n³ + n² -1)/2 squares for n odd.

Unnumbered item at lower right of p. 73. 4 x 4 array of squares with their diagonals drawn, except that the four corner squares have only one diagonal -- the one not pointing to an opposite corner -- and this reduces the number of squares by eight, agreeing with the given answer of 64.

Doubleday - 2. 1971. Bed of nails, pp. 129-130. 20 points in the form of a Greek cross with double-length arms (so that the axes are five times the width of the central square, or the shape is a 9-omino). How many squares can be located on these points? He finds 21.

W. Antony Broomhead. Note 3315: Two unsolved problems. MG 55 (No. 394) (Dec 1971) 438. Find the number of squares on an n x n array of dots, i.e. the second problem in MTg (1965) above, and another problem.

W. Antony Broomhead. Note 3328: Squares in a square lattice. MG 56 (No. 396) (May 1972) 129. Finds there are n²(n²   1)/12 squares and gives a proof due to John Dawes. Editorial note says the problem appears in: M. T. L. Bizley; Probability: An Intermediate Textbook; CUP, 1957, ??NYS. A. J. Finch asks the question for cubes.

Gyles Brandreth. Brandreth's Bedroom Book. Eyre Methuen, London, 1973. Squares, pp. 26 & 63. Same as Briggs.

Nicola Davies. The 2nd Target Book of Fun and Games. 1974. See entry in 5.X.1.

Putnam. Puzzle Fun. 1978.

No. 107: Square the coins, pp. 17 & 40. 20 points in the form of a Greek cross made from five 2 x 2 arrays of points. How many squares -- including skew ones? Gets 21.

No. 108: Unsquaring the coins, pp. 17 & 40. How many points must be removed from the previous pattern in order to leave no squares? Gets 6.
5.X.3. COUNTING HEXAGONS
M. Adams. Puzzle Book. 1939. Prob. C.157: Making hexagons, pp. 163 & 190. The hexagon on the triangular lattice which is two units along each edge contains 8 hexagons. [It is known that the hexagon of side n contains n³ hexagons. I recently discovered this but have found that it is known, though I don't know who discovered it first.]

The Diagram Group. The Family Book of Puzzles. The Leisure Circle Ltd., Wembley, Middlesex, 1984. Problem 32, with Solution at the back of the book. Count the hexagons in the hexagon of side three on the triangular lattice. They get 27.

5.X.4. COUNTING CIRCLES
G. A. Briggs. Puzzle and Humour Book. Published by the author, Ilkley, 1966. Prob. 2/13, pp. 23 & 75. Pattern with hexagonal symmetry and lots of overlapping circles, some incomplete.
5.Y. NUMBER OF ROUTES IN A LATTICE
The common earlier form was to have the route spell a word or phrase from the centre to the boundaries of a diamond. I will call this a word diamond. Sometimes the phrase is a palindrome and one reads to the centre and then back to the edge. See Dudeney, CP, for analysis of the most common cases. I have seen such problems on the surface of a 3 x 3 x 3 cube. The problems of counting Euler or Hamiltonian paths are related questions, but dealt with under 5.E and 5.F.

New section -- in view of the complexity of the examples below, there must be older, easier, versions, but I have only found the few listed below. The first entry gives some ancient lattices, but there is no indication that the number of paths was sought in ancient times.

Roger Millington. The Strange World of the Crossword. M. & J. Hobbs, Walton on Thames, UK, 1974. (This seems to have been retitled: Crossword Puzzles: Their History and Cult for a US ed from Nelson, NY.)

On pp. 38-39 & 162, he gives the cabalistic triangle shown below and says it is thought to have been constructed from the opening letters of the Hebrew words Ab (Father), Ben (Son), Ruach Acadash (Holy Spirit). He then asks how many ways one can read ABRACADABRA in it, though there is no indication that the ancients did this. His answer is 1024 which is correct.

A B R A C A D A B R A

A B R A C A D A B R

A B R A C A D A B

A B R A C A D A

A B R A C A D

A B R A C A