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problem 86 · 8 lines

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<p>A spider, S, sits in one corner of a cuboid room, measuring $6$ by $5$ by $3$, and a fly, F, sits in the opposite corner. By travelling on the surfaces of the room the shortest "straight line" distance from S to F is $10$ and the path is shown on the diagram.</p>
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<img alt="" class="dark_img" src="resources/0086.png"/><br/></div>
<p>However, there are up to three "shortest" path candidates for any given cuboid and the shortest route doesn't always have integer length.</p>
<p>It can be shown that there are exactly $2060$ distinct cuboids, ignoring rotations, with integer dimensions, up to a maximum size of $M$ by $M$ by $M$, for which the shortest route has integer length when $M = 100$. This is the least value of $M$ for which the number of solutions first exceeds two thousand; the number of solutions when $M = 99$ is $1975$.</p>
<p>Find the least value of $M$ such that the number of solutions first exceeds one million.</p>
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