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December 2017 Puzzle Periodical - Reindeer Games!

By Benjamin E., NSA Mathematician

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Can you determine who will win the Reindeer Games?

Problem:

"You know there's Dasher and Dancer and Prancer and Vixen, Comet and Cupid and Donner and Blitzen." But what the song doesn't reveal is that the ordering of the reindeer is determined by how they placed in this year's Reindeer Games, with the winner going first (behind Rudolph, of course) and the last place finisher coming in 8th.

This year, Santa Claus didn't get the full results from the Reindeer Games, since he was busy supervising the elves.

Pencil drawing of reindeer Dasher, Cupid, and Vixen racing, with Vixen in the lead, followed by Cupid, then Dasher. Snowflakes decorate the sky and the title, 'Reindeer Games: Who's Going to win???' is written boldly across the top, adorned with candy canes.Graphics courtesy: Tina M., Captain, USAF

Clues:

Santa has only this information:

  1. Prancer finished three spots behind Dasher, which was better than Blitzen can say;
  2. Donner had his strongest showing ever, but hopes to place higher next year;
  3. Vixen just barely beat out Cupid in the closest finish of the competition;
  4. Comet, Dancer, and Dasher all hoped to finish in the top three; with only Dasher disappointed in the outcome;
  5. Only one reindeer finished between Donner and Dancer.

Reader, with your brain so bright, won't you help Santa sort his reindeer tonight?

Click to see the answer!

Solution:

Pencil drawing of top 3 reindeer on winner's podium, with Dancer in 1st place, Comet in 2nd place, and Donner in 3rd place.Graphics courtesy: Tina M., Captain, USAF

By statement 1, Dasher finished no worse than 4th, as Blitzen is at least 4 spots behind.

By statement 4, Dasher finished no better than 4th. Thus Dasher finished exactly 4th.

Now statement 1 implies Prancer finished 7th and Blitzen must be 8th.

Statement 3 means Vixen finished one spot ahead of Cupid. But from statement 4, Comet and Dancer occupy 2 of the top 3 spots. So the only two adjacent positions remaining are 5 and 6. Therefore, Vixen finished 5th and Cupid was 6th.

We now know the top three finishers include Donner, Dancer, and Comet. Statement 5 implies Comet must come in 2nd place between Donner and Dancer.

Finally, statement 2 implies Donner did not win, so he came in 3rd, leaving the top spot for Dancer!

The Reindeer Games finishers following Rudolph are:

  1. Dancer
  2. Comet
  3. Donner
  4. Dasher
  5. Vixen
  6. Cupid
  7. Prancer
  8. Blitzen

Happy Holidays!