Merry Christmas!
Throughout the month, to accompany the release of our book on the Mathematics of Christmas, Hannah Fry & I were tweeting out Christmathsy bits and pieces, one a day, advent calendar style.
Thanks for following the Indisputable Santa Mathematical Advent Calendar. Just a few loose ends to wrap up, with a solution to our Christmas Eve puzzle today, with the solution to Friday’s puzzle (Santa’s Dressing Room Disasters) to come tomorrow…
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SOLUTIONS TO PREVIOUS PUZZLES
Click here for Saturday’s puzzle.
Scroll down for the solution…
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Solution to Day 24
Let’s number the clues, to make things simpler:
1. Only the front pair contains two reindeer with the same first initial.
2. Cupid is further forward than Dasher.
3. There is exactly one full row between Comet and Blitzen.
4. Prancer is immediately in front of Vixen (nose to tail).
5. Donner is on the same side of the formation as Dasher, but on the opposite side to Dancer.
6. At least one pair of reindeer whose names are listed consecutively in “The Night Before
6. Christmas” are side by side.
7. Fewer than half the four pairs of reindeer are arranged with their two names in alphabetical
7. order (if Santa looks from left to right).
From 1, we know that the front pair are either Comet & Cupid or two of Dasher, Dancer and Donner. However, 2 excludes Dasher. In fact, we can also exclude Comet & Cupid. If they formed the front pair, then 3 implies that Blitzen is in the third row, but then, once you have fitted in Prancer and Vixen nose to tail (4), there is no way to place Dasher, Dancer and Donner without two of them being side by side, which would violate 1. Therefore, the front pair is Donner and Dancer.
Now consider 6. Which pairs of reindeer from the poem could be side by side? Not Dasher and Dancer or Dancer and Prancer, since we’ve already established that Dancer forms a pair with Donner. Not Prancer and Vixen either (by 4). Vixen and Comet is a possibility, but Comet and Cupid is not (by 1). Finally, Cupid and Donner and Donner and Blitzen are impossible, again because Donner has already been paired with Dancer. Therefore, to fulfil 6, Vixen and Comet must be side by side.
Now, since the first row is occupied, 4 implies that Vixen is not in the second row. Also, 3 implies that Comet is not in the third row, and since they are side by side, Vixen and Comet make up the back row. This also means that (by 3) Blitzen is in the second row and (by 4) Prancer is in the third row, immediately in front of Vixen. There are two spaces left, in the second and third rows, and by 2 (and 5), Cupid must be in the second row, with Dasher in the third row (on the same side as Donner).
So, we know that the rows are: Donner and Dancer, Blitzen and Cupid, Dasher and Prancer, Vixen and Comet. We also know that Donner, Dasher and Comet (call them Group 1) are on one side, with Dancer, Prancer and Vixen (Group 2) on the other. We don’t know which of these groups Blitzen and Cupid fit into though.
All this has whittled the possible arrangements down to just four:
- LEFT: Group 1 + Blitzen, RIGHT: Group 2 + Cupid;
- LEFT: Group 1 + Cupid, RIGHT: Group 2 + Blitzen;
- LEFT: Group 2 + Blitzen, RIGHT: Group 1 + Cupid;
- LEFT: Group 2 + Cupid, RIGHT: Group 1 + Blitzen.
Based on everything we’ve learnt, only the final option fits with clue 7 (with Dancer and Donner as the only pair in alphabetical order). Therefore, the true arrangement is:
Dancer , Donner
Cupid , Blitzen
Prancer, Dasher
Vixen, Comet
(Santa in his Sleigh)
Correct answers, this time, came from Olaf Doschke, Troy ToTheWorld, PRL & Catherine Evans. We asked Santa to leave an extra satsuma in your stockings.