Updates to Chapter 8, "Step Right Up!",
in Dissections: Plane & Fancy, by Greg N. Frederickson

Dissecting squares for 22 + 32 + 62 = 72

Jan Kristian Haugland wrote a computer program to generate all 5-piece dissections of a 7-square to a 2-square, a 3-square, and a 6-square. Some of them are quite elaborate. Take a look at his webpage "dissection puzzles".

Fibonacci to Diophantus

Katharina Huber pointed out that the roles of x and y are reversed when reducing Fibonacci's formula to Diophantus's formula on page 73.

Sam Loyd and the 14-15 puzzle

In Sam Loyd's biography on page 73, I wrote that "Loyd popularized the 14-15 puzzle." However, Loyd's claims about having invented the puzzle appear to be false. In a recent book, Jerry Slocum and Dic Sonneveld have documented the origin and spread of the puzzle:
Jerry Slocum and Dic Sonneveld, The 15 Puzzle: How it Drove the World Crazy, Slocum Puzzle Foundation, Beverly Hills, CA, 2006.
As a result of an extensive search in newspapers and magazines from the puzzle's first appearance in 1879, Jerry and Dic concluded that Sam Loyd did not invent the 15-puzzle. (See page 94 of their book for their conclusion, and page 11 to see who probably was responsible.)

Wrong figure referenced

Koji Miyazaki, Hirohisa Hioki, and Naoki Odaka pointed out a problem in the reference in the fourth and fifth lines from the bottom of page 75. It should state that Figure 8.4 closely resembles Figure 7.10, not Figure 7.5.

Penta-penta need not be integral

As suggested by Puzzle 8.3, the Penta class can include identities produced using nonintegral values for p and q. This observation also applies to the Penta-penta class.

Typos in Method 5

There are two typos in the specification of Method 5. In the second line of step 2, the first expression should be y - q - 1 rather than z - q - 1 and the second expression should be y - x rather than z - x.
(These are corrected in the paperback edition.)

Biographical information on Henry Dudeney

More information on Henry Dudeney's life can be found in:
Angela Newing, "Henry Ernest Dudeney, Britain's Greatest Puzzlist", in The Lighter Side of Mathematics: Proceedings of the Eugéne Strens Memorial Conference on Recreational Mathematics and its History, edited by Richard K. Guy and Robert E. Woodrum, pp. 294-301, Mathematical Association of America, 1994.

An earlier dissection

Using Method 2C, I gave 5-piece dissections for all squares in the PP-plus class. One solution in the PP-plus class is 92 + 22 + 62 = 112, for which L. P. Mochalov gave a 5-piece dissection in Vladimir Belov's 1992 book. (See the updates to chapter 1.) Mochalov's method can be generalized to handle all squares in the PP-plus class.

Cossali's name

I should have written Cossali's name as Pietro Cossali.

Another way to generate Cossali's class

There is another way to produce Cossali's class without resorting to the device of n=m or n=p, as I did in the book. Instead, take p, q, and n all to be 1, and take m to to be odd. This will give values that have a common factor of 4. The values are then in a different order from what you see on page 84.

Another class of 3 squares to 1

David Eppstein (at the University of California, Irvine) has pointed out that Robert Reid's 5-piece dissection of 4-square, 9-square, and 48-square to a 49-square in Fig. 8.19 is not an isolated dissection. Instead, it belongs to a class whose first four members are (1, 2, 2, 3), (4, 9, 48, 49), (21, 50, 1470, 1471), and (120, 289, 48960, 48961).
Members of the class are integral solutions to the simultaneous equations:
x (z + 1 - y - x) = (x - 1) z
z +
1 - y = (y - 1) (y - x)
x2 + y2 = 2 z +
1
w = z +
1
All solutions to these equations can be found by the following method. Consider a series of pairs of values ai,bi satisfying:
a1 = b1 = 1
ai+1 = ai + bi
bi+1 = 2ai + bi
We can prove by induction that:
(b2j)2 = 2(a2j)2 + 1
(b2j+1)2 + 1 = 2(a2j+1)2
Let y2j = (b2j)2 and y2j+1 = (b2j+1)2 + 1. Then take xi = sqrt(yi-1(yi-1)).
It follows that zi = xi sqrt(2yi(yi-1)).
Thus the values of xi, yi, zi, and wi satisfy the requirements for Reid's dissection.
Note that bi / ai is an approximation for the square root of 2, which is generated by the continued fraction technique for solving the Pell equation b2 + 1 = 2a2. About 130 A.D. Theon of Smyrna gave the recursive formulas for ai and bi. They had earlier been given in geometric form. (See page 341 of Dickson (1920).) Thus I will call this class of solutions to x2 + y2 + z2 = w2 Theon's class.
Since the 5-piece dissection for this class uses two steps, and the steps are specializations of P-slides, the existence of this class suggests a relationship for three squares to one that I missed in chapter 4. Squares for that relationship will have 7-piece dissections.

n Squares to one

I identified many dissections of three squares to one that use just five pieces, and also gave one dissection of four squares that uses six pieces. In 1978, A.J.W. Duivestijn published a set of 21 squares of different sizes that can be rearranged without cutting to form a single square. This leaves open the problem of dissecting n different-sized squares, for n between 3 and 21, into the fewest number of pieces that form a larger square. Another optimization criterion is to minimize the total length of the cut. Joe Kingston and Des MacHale investigate this latter problem in their article, "Dissecting Squares," which is to appear in the Mathematical Gazette in November 2001. Since submitting their article, they have found some improved results, which they describe in "Some Improved Dissections of Squares."


Copyright 1997-2017, Greg N. Frederickson.
Permission is granted to any purchaser of Dissections: Plane & Fancy to print out a copy of this page for his or her own personal use.

Last updated May 3, 2017.