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How many parallelograms can be found in a equilateral triangle of length 10 units divided equally into equilateral triangle of length 1 unit?

Can anyone come up with a detailed solution?

user168802
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    I find necklace problems the hardest to solve...don't know what that implies about olympiads. – Oria Gruber Sep 18 '14 at 11:29
  • Can I have an example? – user168802 Sep 18 '14 at 11:31
  • I've found the famous 'Hat Check Problem' to be quite very difficult. Here are a couple of examples asked on this forum: http://math.stackexchange.com/q/876126/131263 and http://math.stackexchange.com/q/627913/131263. In addition to that, Integer Partitioning questions can also be very difficult (see http://en.wikipedia.org/wiki/Partition_%28number_theory%29). – barak manos Sep 18 '14 at 11:39
  • @user168802 : How many necklaces can be formed by using 3 white pearls, 4 red pearls, and 5 green pearls? Studying under Noga Alon made me quite efficient at solving problems like these :) – Oria Gruber Sep 18 '14 at 12:02
  • Thanks you so much! Oria Gruber, Barak Manos. – user168802 Sep 18 '14 at 12:17
  • Integer Partitioning is quite interesting! – user168802 Sep 18 '14 at 12:19
  • Okay, how about this one for a bit? How many ways can you mark 8 squares of an 8 by 8 chessboard so that no two marked squares are in the same row or column, and none of the four corner squares is marked? (Rotations and reflections are considered different) – user168802 Sep 18 '14 at 12:20

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Extend the triangle down one more row. Then extend the sides of the parallelogram until they hit that row, and mark those spots they hit.

The bijection should now be clear.

extremeaxe5
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