<s>[INST] Finding the maximum value of $n$ What's the maximum value for $n$ for which there is a set of distinct positive integers $k_{1}, k_{2}, ... , k_{n}$ for which $k_{1}^{2} + k_{2}^{2} + ... + k_{n}^{2} = 2002$
 [/INST] $1^2+2^2+3^2+4^2+5^2+6^2+7^2+8^2+9^2+10^2+11^2+12^2+14^2+16^2+18^2+24^2=2002$ which uses 16 distinct squares.
$1^2+\dots+18^2=2109>2002$, so any 18 distinct squares have too large a sum. So the only question is whether it can be done with 17.
Note that $1^2+2^2+\dots+17^2=1785=2002-217$ and $18^2-10^2=224>217$, so we must include $1^2+2^2+\dots+10^2=385$ if we have 17 squares.
If we start from $1^2+\dots+17^2=1785$, and add a square larger than $18^2$ we are adding at least $19^2=361$, so we must drop at least $12^2$. That works!
$1^2+\dots+11^2+13^2+14^2+15^2+16^2+17^2+19^2=2002$ with 17 squares.
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