[vc_row][vc_column][vc_column_text]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?[/vc_column_text][vc_toggle title="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 $16\textrm{m}\times16\textrm{m}\times16\textrm{m}$. [latexpage]
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
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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)
