Sled-packing with Santa


Santa has packed up all his presents into 16 boxes. Each box is a cube: the first is 1m×1m×1m, the second 2m×2m×2m, the third 3m×3m×3m, … up to the largest which is 16m×16m×16m. He wants to ft all the boxes onto two sleds but is bound by the Elf Safety Regulations. These state that on each sled he must have the same number of boxes. In addition to this on each sled the sum of the side-lengths of the boxes must be the same, the sum of the areas of the bases of the boxes must be the same and the sum of the volumes of the boxes must be the same.

Is this possible?

Reveal Solution

Santa has packed up all his presents into 16 boxes. Each box is a cube: the first is 1\textrm{m}\times1\textrm{m}\times1\textrm{m}, the second 2\textrm{m}\times2\textrm{m}\times2\textrm{m}, the third 3\textrm{m}\times3\textrm{m}\times3\textrm{m}, … up to the largest which is 16\textrm{m}\times16\textrm{m}\times16\textrm{m}.

He wants to fit all the boxes onto two sleds but is bound by the Elf Safety Regulations. These state that on each sled he must have the same number of boxes. In addition to this on each sled the sum of the side-lengths of the boxes must be the same, the sum of the areas of the bases of the boxes must be the same and the sum of the volumes of the boxes must be the same. Is this possible?

One possible solution to this puzzle is

\boldsymbol{n} \boldsymbol{n}^2 \boldsymbol{n}^3 \boldsymbol{n} \boldsymbol{n}^2 \boldsymbol{n}^3
1 1 1 2 4 8
4 16 64 3 9 27
6 36 216 5 25 125
7 49 343 8 64 512
10 100 1000 9 81 729
11 121 1331 12 144 1728
13 169 2197 14 196 2744
16 256 4096 15 225 3375
Total 68 748 9248   68 748 9248
Problem Page Coordinator: Claire Baldwin – Mathematics in Education and Industry
Acknowledgement: The IMA are indebted to MEI for sourcing and supplying Mathematics Today with these well-known puzzles.
First published in Mathematics Today (December 2017)
Image credit: Cheerful happy beautiful elf is holding two Christmas Gifts by © Disarny / Dreamstime.com
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