Selling Sheep

My National Review Online "Diary" column for May 2003 included a brainteaser about two brothers who sold their sheep at market.

Selling Sheep

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Brothers Angus and Hughie raise sheep in the Highlands.  They take the flock to market one day and sell all the sheep.  By odd coincidence, the price per sheep was the same as the number of sheep, that is, the total sum received by the brothers was a perfect square.

The brothers collect their earnings as a stack of 10-pound notes and less than ten 1-pound coins.  They go home with the intention to divide it 50-50.  The ten-spots are counted out, but Hughie winds up with one more ten-spot than Angus. Hughie shoves the stack of coins across the table to Angus and says:  "That makes it close to even, but not quite. I'll write you a draft to square our accounts."

How much was the draft for?

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The solution hinges on the following fact about perfect squares.  (That is, the squares of whole numbers.)

A perfect square that has an odd number of tens must end in 6.

This is because, if you square a whole number, your answer is (a) an even number of tens, plus (b) the square of your number's units.  Examples:

Square 12.  You get 140 (that is, fourteen tens) plus 4 (the square of 2).

Square 37.  You get 1320 (that is, 132 tens) plus 49 (the square of 7).

Square 7,239.  You get 52,403,040 (that is, 5,240,304 tens) plus 81 (the square of 9).

This is bound to be the case.  If the number of tens in your original number is T, and the number of units is U, then your number is 10T+U.  Squaring that gives you 100*T² + 20*T*U + U².  The number of tens in this answer is 10*T² + 2*T*U, which is certainly an even number:  it is 2 times the whole number  5*T² + T*U.

The only way a perfect square can have an odd number of tens, therefore, is by acquiring them when you add U².  If I add something to an even number and get an odd result, the thing I added must have been odd.  The number of tens in U² therefore needs to be odd.  Since U is a single digit from 0 to 9, U² is one of the following:  0, 1, 4, 9, 16, 25, 36, 49, 64, 81.  The only ones with an odd number of tens are 16 and 36.  Q.E.D.

Now, since the total sum the brothers are dividing is a perfect square, and since we know that the number of tens is odd, we know that the number of units is 6.  In bills, Hughie has one more 10-spot than Angus.  After giving him the 6 coins, he is up by only 4 pounds.  

Therefore the draft is for two pounds.