Inheriting a Flock of Sheep

[Rosenfeld]

Two brothers inherit a flock of sheep. Not being shepherds, they decide to sell the sheep and divide evenly the money that they receive for the flock. They sell the entire flock, receiving as many dollars per sheep as there are sheep in the flock. They are paid in 10 dollar bills and the remainder (less than 10), in one dollar bills. The older brother takes the money, and sitting down at a table with the younger brother, divides the money.

Starting with the 10 dollar bills, he counts " 10 for me, 10 for you, 10 for me, 10 for you, ......., 10 for me (with the last 10 dollar bill)." The younger brother complained, "You took the first bill and the last bill. You received an extra 10 dollars." "You're right", said the older brother, "take all of the 1's". "That's still not enough", said the younger brother. "That's right again", said the older brother. So, to be fair, the older brother gave the younger one a check for half of the difference.

For what amount was the check?

------
Irv

[hyh1048576]

1,right?
first,n^2(mod10)<5("still not enough").So it can only be 1 or 4.And "take all of the 1's" suggest that it's 4.So the difference is 2. :)
6-28，19：31—— ops: ops: ops: sorry,Wackerbarth is right :( :( :(

[Wacky]

Taking N as the number of Sheep.
The total received is N^2.
They evenly divide (N^2)/20 and the remainder is (N^2)mod20.
Since the first gets 10 and the second gets the rest, this is unfair only if
(N^2)mod20 > 10.
That condition holds only for Nmod10 = +/- 4. Or Nmod10 = 4 or 6.
Then (N^2)mod20 = 16. So half of that is 8.
The first brother received 10 and the second received only 6.
The check is for 2.
10 - 2 = 8 = 6 + 2