When my cats aren't happy I'm not happy. Not because
I care about their mood, but because I know they're
just sitting there thinking up ways to get even.
~Penny Ward Moser
If Cabe and Dave each see three different numbers,
and if Babe can see number 2 twice, this number must
be facing up.
If Abe sees two even numbers, plus the number 1, it
follows that he is looking at 1, 2, and 2 (from left
to right).
Dave, on the other hand, looks at the numbers 3, 2,
and 1, Cabe at the numbers 1, 2, and 3, and Babe can
see 2, 2, and 1.
Therefore the number hidden from view is 3.
The only lockers that remain open are perfect squares
(1, 4, 9, 16, etc) because they are the only numbers
divisible by an odd number of whole numbers; every
factor other than the number's square root is paired
up with another. Thus, these lockers will be
"changed" an odd number of times, which means they
will be left open. All the other numbers are
divisible by an even number of factors and will
consequently end up closed.
So the number of open lockers is the number of
perfect squares less than or equal to one thousand.
These numbers are one squared, two squared, three
squared, four squared, and so on, up to thirty one
squared. (Thirty two squared is greater than one
thousand, and therefore out of range.) So the answer
is thirty one.
50 minutes. The game lasts 75 minutes and 12 players can be in the game for its duration, so there are a total of 900 player minutes (75 X 12). If 18 players are involved, the sum is 900 ÷ 18 = 50.