Problem 6

Prove that there exists a convex 1990-gon whose angles are all equal and whose side lengths are

1^2,2^2,\ldots,1990^2

in some order.

Step 3 of 4: Match each direction with a distinct squared side length

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r(i,j,k)=1+995i+199j+k,\qquad \ell_{i,j,k}=r(i,j,k)^2

Detailed analysis

Define $r(i,j,k)=1+995i+199j+k$ . As $0\le i<2$ , $0\le j<5$ , and $0\le k<199$ , these values run bijectively through $1,\ldots,1990$ . Assign the squared length $\ell_{i,j,k}=r(i,j,k)^2$ to the direction $(-1)^ia^jb^k$ ; thus every required side length is used exactly once.