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GeometryQuestion and Answers: Page 8

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Given is a square with side length 15. We need to find exactly 17 smaller squares to fill the big one. How many solutions are possible? (Note: it′s not enough to find squares with the sum of their areas being 225, they must fit into the 15×15 square. Example with 3 squares: 2×2+5×5+14×14=225 but you cannot fit these in a 15×15 square)

$$\mathrm{Given}\:\mathrm{is}\:\mathrm{a}\:\mathrm{square}\:\mathrm{with}\:\mathrm{side}\:\mathrm{length}\:\mathrm{15}. \\ $$$$\mathrm{We}\:\mathrm{need}\:\mathrm{to}\:\mathrm{find}\:\mathrm{exactly}\:\mathrm{17}\:\mathrm{smaller}\:\mathrm{squares} \\ $$$$\mathrm{to}\:\mathrm{fill}\:\mathrm{the}\:\mathrm{big}\:\mathrm{one}.\:\mathrm{How}\:\mathrm{many}\:\mathrm{solutions}\:\mathrm{are} \\ $$$$\mathrm{possible}? \\ $$$$\left(\mathrm{Note}:\:\mathrm{it}'\mathrm{s}\:\mathrm{not}\:\mathrm{enough}\:\mathrm{to}\:\mathrm{find}\:\mathrm{squares}\:\mathrm{with}\right. \\ $$$$\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{their}\:\mathrm{areas}\:\mathrm{being}\:\mathrm{225},\:\mathrm{they}\:\mathrm{must} \\ $$$$\mathrm{fit}\:\mathrm{into}\:\mathrm{the}\:\mathrm{15}×\mathrm{15}\:\mathrm{square}.\:\mathrm{Example}\:\mathrm{with} \\ $$$$\mathrm{3}\:\mathrm{squares}:\:\mathrm{2}×\mathrm{2}+\mathrm{5}×\mathrm{5}+\mathrm{14}×\mathrm{14}=\mathrm{225}\:\mathrm{but} \\ $$$$\left.\mathrm{you}\:\mathrm{cannot}\:\mathrm{fit}\:\mathrm{these}\:\mathrm{in}\:\mathrm{a}\:\mathrm{15}×\mathrm{15}\:\mathrm{square}\right) \\ $$

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