MATHEMATISCHE ZEITSCHRIFT, 45 (1939), pp. 607–608.

An example of a dissection of the square into pairwise unequal

squares. by R. Sprague in Berlin-Charlottenburg.

Herr Stöhr^{1} was recently occupied with the question of whether a rectangle of given aspect ratio, in particular a square, can be dissected into finitely many pairwise different squares. Theorem 10 of his dissertation states:

If any rectangle with sides a and b, a < b, admits two such dissections, which have no component square in common, and do not contain the square with side a, then the square with side a + b can be dissected in the desired way (namely, into the squares with sides a and b and the two given dissected rectangles.)

In the following we give two dissections of a rectangle with aspect ratio 13 : 16 which satisfy the conditions of this theorem.

The first is shown in Figure 1; it has been based on well-known^{2} dissections of the rectangles with aspect ratio 32 : 33 and 47 : 65 into pairwise unequal squares, which coincidentally have no square in common. The numbers denote the side lengths of the component squares.

We name the rectangle of Fig. 1 R_{1}; it has sides of 5 · 13 and 5 · 16 and it does not contain the square with side equal to its smaller side as a component.

In order to find the second decomposition, we first augment R_{1} with a square along its longer side, thus producing a rectangle R_{2} with sides 5 · 16 and 5 · 29. Furthermore, a rectangle R_{3} with sides 16 · 16 and 13 · 29 is used; its dissection is shown in Figure 2.

The sides of the squares are

in R_{2}: 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 17, 18, 19, 22, 23, 24, 25, 33, 80;

in R_{3}: 7, 10, 28, 54, 61, 68, 75, 113, 115, 123, 133, 141.

1A. Stöhr, Über Zerlegungen von Rechtecken in inkongruente Quadrate. (Dissertation.)

Schriften des Math. Inst. und des Inst. f. angew. Math. der Universität Berlin, Band 4 (1939), S. 119–140. Vgl. die Literaturangaben auf S. 119 und 120.

2Jaremkewycz, Mahrenholz, Sprague, Lösungen der Aufgabe 1242. Zeitsch. f. d. math.

u. naturwiss. Unterricht 68 (1937), S. 43.

Linear enlargement of R_{2} in the ratio 1 : 13 and of R_{3} in the ratio 1 : 5 yields rectangles R′_{2} with sides 5 · 13 · 16 and 5 · 13 · 29 and R′_{3} with sides 5 · 16 · 16 and 5 · 13 · 29.

Joining R′_{2} and R′_{3} at the sides of equal length creates one rectangle R_{4} with sides 5 · 13 · 29 and 5 · 16 · 29. It does not contain the square with side equal to its smaller side as a component. The rectangle R_{4} contains pairwise unequal squares. This is because R_{2} and R_{3} have this property, and unlike the squares in R′_{2}, no square in R_{3}, and hence none in R′_{3}, has side a multiple of 13.

No side of a square in R_{2} or R_{3} is a multiple of 29; therefore, the same is true for R_{4}. The enlargement of R_{1} in the ratio 1 : 29 and R_{4} thus are two dissections of the rectangle with sides a = 5 · 13 · 29 and b = 5 · 16 · 29 which fulfill the conditions.

R_{1} consists of 20, R_{4} of 33 component squares; two more component squares also appear, those with sides a and b. Therefore, each square is dissectable into 55 pairwise unequal squares.