// You want to show 3^3 + 4^3 + 5^3 = 6^3 (because of that) // but you've forgotten how to multiply integers together. -- 6^3 = (3+3)^3 = 3^3 + 3·3^3 + 3·3^3 + 3^3 = 8·3^3 5^3 = (3+2)^3 = 3^3 + 3·3^2·2 + 3·3·2^2 + 2^3 = 3·3^3 + 2^2·3^2 + 2^3 4^3 = (3+1)^3 = 3^3 + 3·3^2 + 3·3 + 1 = 2·3^3 + 3·3 + 1 --- 3^3 + 4^3 + 5^3 = 3^3 + 2·3^3 + 3·3 + 1 + 3·3^3 + 2^2·3^2 + 2^3 = 6·3^3 + 3^2 + 1 + 2^2·3^2 + 2^3 // Now we need to show that 3^2 + 1 + 2^2·3^2 + 2^3 = 2·3^3 // Note that 2^2 = (1+1)^2 = 1 + 2 + 1 3^2 + 1 + 2^2·3^2 + 2^3 = 3^2 + 1 + (1+2+1)·3^2 + 2^3 = 3^2 + 2·3^2 + 2·3^2 + 1 + (1+2+1)·2 = 3·3^2 + 2·3^2 + 1 + (1+2+1)·2 = 3^3 + 2·3^2 + 1 + (1+2+1)·2 // so now we just need 2·3^2 + 1 + (1+2+1)·2 = 3^3 - 2·3^2 + 1 + (1+2+1)·2 = 2·3^2 + (3+1)·2 + 1 = 2·3^2 + 3·2 + 1·2 + 1 = 2·3^2 + 3·2 + 1·(2 + 1) = 2·3^2 + 3·2 + 1·3 = 2·3^2 + 3·3 = 2·3^2 + 3^2 = 3·3^2 = 3^3 // Tada!