This was a letter I wrote to the editor of
*Journal of Recreational Mathematics*.
It was published in volume 25 (1993), no. 2, on pages 149-150.

Joseph S. Madachy

Editor, Journal of Recreational Mathematics

In a Letter to the Editor [1], Jean Meeus asks for the smallest rectangle that may be tiled by the following pairs of pentominoes: IF , IT , IU , IW , IX , IZ . Originally, I found by hand rectangles of sizes 11 x 20 , 10 x 11 , 13 x 20 , 11 x 20 and 16 x 50 for the pairs IF , IT , IU , IW and IZ , respectively. However, with the use of a computer, I was able to find the smallest rectangles for each of these pairs. The 10 x 11 IT rectangle was minimal, but for IF , IU , IW and IZ , the minimal rectangles have sizes 10 x 11 , 12 x 20 , 12 x 15 and 10 x 16 . The 10 x 11 IF rectangle has already been given by Rodolfo Kurchan [2]. The others are illustrated. Note that the IZ pair also tiles an 11 x 15 rectangle, so that a 15 x 15 square is the smallest IZ square. I suspect that an IX rectangle is not possible.

References:

[1] Jean Meeus, Letter to the Editor, *Journal of Recreational
Mathematics* **18** (1985-1986), no. 1, pp. 49, 51.

[2] Rodolfo Marcelo Kurchan, Letter to the Editor, *Journal of
Recreational Mathematics* **23** (1991), no. 1, p. 5.

IT 10 x 11

IU 12 x 20

IW 12 x 15

IZ 10 x 16

IZ 11 x 15

Although I did not include them with the letter, it may be of interest to see the tilings I found manually, so I include them here. (The IT rectangle is above.)

IF 11 x 20

IU 13 x 20

IW 11 x 20

IZ 16 x 50

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Updated May 15, 2005.