Coonamble has had all this rain, and there is a terrible mouse plague. Dead and dying mouses everywhere, some with blue teeth.

Referring to my previous two solutions, here is a third and better way to do the original problem: 14^2 - 2^2 = 12*16 (difference of squares) 13^2 - 1^2 = 12*14 12^2 - 0^2 = 12*12 11^2 - (-1)^2 = 12*10 10^2 - (-2)^2 = 12*8 Adding the five lines: (sum of squares from 10^2 to 14^2) = 5*12^2 + (sum of squares from (-2)^2 to 2^2) This allows three more general problems to be investigated: PROBLEM 1: Find all sums of five consecutive squares divisible by 365, and find the resulting quotients. PROBLEM 2: Find all sums of (2n+1) consecutive squares divisible by 365, and find the resulting quotients. PROBLEM 3: This year 2020 is a leap year. Replace 365 = 5*73 by 366 = 6*61, then by any fixed whole number.

Use squares of binomials: 10^2 = (12 - 2)^2 = 12^2 - 4*12 + 4 14^2 = (12 + 2)^2 = 12^2 + 4*12 + 4 11^2 = (12 - 1)^2 = 12^2 - 2*12 + 1 13^2 = (12 + 1)^2 = 12^2 + 2*12 + 1 12^2 = 12^2 Add them up, sum = 5*(12^2) + 5*2 = 5 * 146 Denominator = 365 = 5 * 73 Hence ratio = 146/73 = 2 The difference of squares is quicker: (14^2 - 12^2) + (10^2 -12^2) = 26*2 - 22*2 = 4*2 (13^2 - 12^2) + (11^2 -12^2) = 25*1 - 23*1 = 2*1 Hence 10^2 + 11^2 + 12^2 + 13^2 + 14^2 = 5*12^2 + 5*2 = 5 * 146 But this year is a leap year!