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10 + 11 + 12 + 13 + 14 + 15 + 16 & = & 27 + 64

Source: Polish MO second round 1953 p1

August 29, 2024
algebraSum

Problem Statement

The board was placed
<br/>1=1<br/>2+3+4=1+8<br/>5+6+7+8+9=8+27<br/>10+11+12+13+14+15+16=27+64<br/>…<br/> \begin{array}{rcl}<br /> 1 & = & 1 \\<br /> 2 + 3 + 4 & = & 1 + 8 \\<br /> 5 + 6 + 7 + 8 + 9 & = & 8 + 27\\<br /> 10 + 11 + 12 + 13 + 14 + 15 + 16 & = & 27 + 64\\<br /> & \ldots &<br /> \end{array} Write such a formula for the n n -th row of the array that, with the substitutions n=1,2,3,4 n = 1, 2, 3, 4 , would give the above four lines of the array and would be true for every natural n n .
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