A sequence of triangles is constructed in the following way:

1) the first triangle is Pythagoras' 3-4-5 triangle

2) the second triangle is a right-angle triangle whose second longest side is the hypotenuse of the first triangle, and whose shortest side is the same length as the second longest side of the first triangle

3) the third triangle is a right-angle triangle whose second longest side is the hypotenuse of the second triangle, and whose shortest side is the same length as the second longest side of the second triangle etc.

Each triangle in the sequence is constructed so that its second longest side is the hypotenuse of the previous triangle and its shortest side is the same length as the second longest side of the previous triangle.

What is the area of a square whose side is the hypotenuse of the nth triangle in the sequence?

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10 older comments

Jean
on 12 Dec 2013

good for brain !

Jean-Marie Sainthillier
on 13 Dec 2013

The best problem of the CUP challenge.

Jean-Marie Sainthillier
on 13 Dec 2013

I think that thematic challenge is a wonderful idea. I solved this one with a great pleasure.

Matt
on 4 Feb 2014

Great problem. So many ways to solve it. It was fun trying different methods to see which scored the best.

kwijibo28
on 8 Feb 2014

The connection of this problem with the Fibonacci sequence is most interesting.

Philipp
on 5 Mar 2014

indeed: like!

Guilherme Coco Beltramini
on 26 Jun 2014

Very interesting problem!

Vishal
on 8 Sep 2014

Interesting problem....!

Vidushi Jain
on 10 Jun 2015

Interesting!!

Pooja Narayan
on 23 Feb 2016

Good one :)

Giovanni Mottola
on 7 Jun 2016

Combining the Fibonacci sequence and the Pythagorean theorem is a nice idea

Dhaval
on 11 Sep 2016

I thought I had the right code and it didn't work. Then I realized that the area of the first triangle (n=1 case) should be 0.5*4*3 = 6. But is given in the solution as 25. Am I understanding this wrong? I thought the three sides of the first triangle (n=1) was 3,4,5 and we go on from there. Please let me know if my understanding is wrong and then I can proceed Otherwise I have a general code.

Milan Petrovic
on 21 Oct 2016 at 9:41

@ Dhaval: Try reading the last sentence of the problem very carefully.

1 Comment

Thomas Blackwood
on 7 Sep 2016

If anyone cares to point out why this is wrong, I'm open to hearing.

1 Comment

John D'Errico
on 21 Aug 2016

Note that the ratio of consecutive areas is asymptotically the golden ratio, i.e., phi=(1+sqrt(5))/2.

1 Comment

Daniel Zimmermann
on 16 Jul 2016

Weak! My recursion solution should work just fine, but the server is unable to evaluate it properly. I'm going to count it.

1 Comment

Carlos Zúñiga
on 9 Mar 2016

Its the best way that I found :)

1 Comment

Raphael
on 20 Mar 2014

I consider my answer as a cheat

1 Comment

Jon
on 3 Feb 2014

yes, this is probably cheating

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36 players like this problem

36 players like this problem