Enlarging a square orchard

MediumBrainteasers~6m

A gardener plants trees in a square array and has $146$ trees left over. To make the square one row larger in each direction, he would need $31$ more trees than he has. How many trees does he have in total?

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Side $k\to k+1$ needs $2k+1$ more trees. That equals surplus $+$ shortfall $=146+31$ .
$2k+1=177\Rightarrow k=88$ ; total $=88^2+146$ .

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Enlarging a square orchard

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