Sam decided to divide his cornfield between himself and his four sons Al, Bert, Connor and David in proportion to their work rates. He knew that Al, Bert, Connor and David together could plant a field of corn in five hours - whereas if Sam replaced Al, the same task would take six hours.

Accordingly, Sam divided his field into a number of equal parts equal to the plot number of the cornfield. He kept one part for himself, and gave the same whole number of parts to Bert, Connor and David, giving the other parts to Al. (The plot number of the cornfield in base 10 is a square number between 10 and 99 inclusively.) How many parts did each son get?

Let A, B, C, D and S be the allocations to the five people.

What we are told about the allocations and about what happens when Sam replaces Al,

A + B + C + D = 6k
while
S + B + C + D = 5k

We are told

S = 1

so

A - S = A - 1 = k, and therefore k is an integer.

Also,

B = C = D = (5k - 1) / 3

remembering that A = k + 1 and S = 1.

Trying different values of k that lead to integral B, C, D:

  k |  B   C   D   A   S | Tot
 ---+--------------------+---- 
  2 |  3   3   3   3   1 |  13
  5 |  8   8   8   6   1 |  31
  8 | 13  13  13   9   1 |  49  which is a perfect square
 11 | 18  18  18  12   1 |  67
 14 | 23  23  23  15   1 |  85
 17 | 28  28  28  18   1 | 103  now too large 
 
 So B, C and D each received 13 parts and A got 9.

Posted by Charlie on 2008-12-31 15:39:29