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I was going through this question given to my friend at university, and I can do it with a little calculation. Is there a set algorithm that will work or a step-by-step guide that we can use so that you can solve this question for different value like $9$ and $10$?

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1 Answer

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Suppose the question asks for each side of the triangle summing up to $c$.

Then the sum of all three sides, counting each corner twice, is $3c$.

By this requirement, the sum of numbers in the corners is $3c-1-2-3-4-5-6 = 3c-21$.

For example, consider part (c), where $c = 11$.

Then the corner numbers sum to $3\times11-21=12$.

The three numbers, taken from $1$ to $6$, must be one of the triples:$$(3,4,5),(1,5,6),(2,4,6)$$

Which number belongs to which corner does not matter by symmetry.

By $3+4+6>11$ and $5+6=11$ we conclude the first two triples are invalid.

Thus the corner numbers must be $2, 4, 6$.

Finally it's a simple process to fill in the rest of the numbers.

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