**From New Scientist #2871, 30th June 2012** [link]

Joe drew a right-angled triangle on an A6 file card. He asked Penny to cut it out and then cut it in two to make two right-angled triangles. Then he asked her to cut each triangle in two to make four right-angled triangles in total. Now Joe’s triangle was very special. Penny found that the lengths of all the sides of all the triangles were a whole number of millimetres.

What was the area of the smallest triangle?

**Note:** An A6 card has dimensions of 105 mm × 148 mm.

[enigma1704]

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The following Python program runs in 50ms.

Solution:The area of the smallest triangle is 486 sq mm.Here’s a diagram showing the solution:

I’m not sure if the intention is to use the whole of the card in some way, but if not there are multiple solutions.

No there aren’t. I missed one of the equations from the code:

More understandable code, and a picture:

Here is my solution:

Brian’s observation that all the triangles are similar is the key to a simple solution for non-coders. This diagrams illustrates the case of a 3,4,5 triangle scaled up. https://www.dropbox.com/s/ip38qog0gy9xpne/Enigma%201704%20a.PNG

The bottom side is 125n/12, so n must be 12.

I’m guessing that a similar exercise on the next triangle, 5,12,13 will exceed the limits of A6.

But the answer is a whole number of millimetres. And A6 is 105mmx148mm.

There is definitely room for it I think.

Hi Claire-Louise,

Yes, but it also has to have the four integer sided right angled angles triangles inside it and this is not possible