## Brain-Teaser 494

**From The Sunday Times, 15th November 1970** [link]

Tina and Cindy each started with a rectangular sheet of cardboard. The two rectangles had equal dimensions.

Each girl cut her sheet into 5 squares using 4 straight cuts; the first 3 cuts producing one square each and the fourth cut producing two squares. The first square obtained by Tina had twice the area of the first square obtained by Cindy.

If Tina’s fifth square had an area of one square inch, what was the area of the rectangles?

[teaser494]

## Jim Randell 3:24 pm

on18 August 2019 Permalink |Instead of considering cutting up the rectangle we consider adding squares to an existing rectangle.

Tina’s 4th and 5th squares are 1 inch squares, so come from a 2 × 1 rectangle. A square can be added to any rectangle by extending it horizontally or vertically, and we need to add three more squares, so there are 8 possible starting rectangles.

Cindy’s rectangle is also one of these 8 appropriately scaled to make the rectangles have the same area. If Cindy’s rectangle is scaled by

falong each side then the area is scaled byf².Also the area of Tina’s first square is twice the area of Cindy’s first square.

This Python 3 program finds the 8 possible rectangles and their divisions and finds a pair that give an appropriate scale factor.

It runs in 87ms.

Run:[ @repl.it ]Solution:The area of the rectangles was 28 square inches.Tina’s rectangle was 4×7, and her squares had area: 16, 9, 1, 1, 1.

Cindy’s rectangle was (7√2)×(2√2), and her squares had area: 8, 8, 8, 2, 2.

Here is a diagram of the two rectangles:

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