MEI Maths Item of the Month
December 2015
The Elf Lottery
Santa has 100 reindeer in 100 numbered stalls and 100 elves to boss about. He picks
reindeer by the following system:
Initially all the stall doors are closed. Elf 1 opens every door, elf 2 then goes to every other
door (i.e. multiples of 2) and changes its state (open to closed or vice versa), elf 3 does this
for every multiple of 3 door and so on to elf 100.
On Christmas Eve Santa chooses the reindeer in stalls with open doors. Which reindeer
does Santa choose?
Solution
The square numbered doors will be left open and Santa will choose the reindeers in these
stalls.
Each door will be either opened or closed by every elf whose number is a factor of the door
number. All numbers have an even number of factors except square numbers. The nonsquare numbered doors will be open/closed an even number of times and hence result in
being closed. The square numbered doors will be open/closed an odd number of times and
hence result in being open.
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