The diagram below illustrates one large equilateral triangle composed of smaller equilateral triangles. The number shown within each triangle is the length of one side relative to the other triangles. Every triangle must have a non-zero integer value. If we doubled the sides of every triangle we would get the same pattern but with double the numerical values; the triangle shown is thus also the smallest possible with that particular tiling.

The

*order*of such a triangle is the number of smaller triangles – in this case, the order is 8.

The

*size*of the triangle is the length of the large triangle – in this case, the size is 5.

The

*sum*of the triangle is the sum of the numbers within the smaller triangles. However, the rule to calculate the sum is that triangles pointing upwards are positive, while those pointing downwards are negative. So in this case, the sum is equal to 3 + 2 + 2 + 1 + 1 – 2 – 1 – 1 = 5.

Is the sum of such a triangular tiling always equal to its size?

**Preliminary Question**

There are only 3 triangle tilings of order 8; one of them is the one in the diagram. Find the other two tilings. You may write your answer as a list of the triangle sizes. Tilings that are either rotations or reflections of each other count as just one unique solution.

**The Real Question**

Before spelling out the full question, one last definition is required. A tiling is said to be a

*prime tiling*if all the constituent smaller triangles have different numbers. The numbers themselves do not have to be prime or relatively prime to other numbers – they just have to be unique within the tiling. Also, triangles with the same side length but opposite signs are counted as different numbers.

So, a prime triangle tiling of order N (N>1) will have N triangles, none of which will have the same integer size. The triangle with N=1 shall be our basic unit triangle.

The smallest possible prime triangle tiling has an order of 15 and a large triangle of size 39. There are two such triangles. Find one of them and calculate its sum. Instead of sending an image, you may submit your full calculation for the sum, including all the individual triangle sizes.

Have fun!

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