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Solution to puzzle 35: Cuboids

Consider a cuboid of dimensions a × b × c, where a less than or equal to b less than or equal to c.
The total number of unit cubes is abc.
The total number of internal unit cubes is (a − 2)(b − 2)(c − 2).

Hence we seek positive integers a less than or equal to b less than or equal to c such that abc = 2(a − 2)(b − 2)(c − 2).(1)

Dividing by abc, we obtain (1 − 2/a)(1 − 2/b)(1 − 2/c) = ½.
Since (1 − 2/a) less than or equal to (1 − 2/b) less than or equal to (1 − 2/c), we have 1 − 2/a less than or equal to cuberoot(½), and so a less than or equal to 2/(1 − cuberoot(½)).
Therefore a less than or equal to 9.
This places a precise bound on the intuition that a cuboid with equal numbers of internal and external cubes cannot be "too large."

Expanding (1), we obtain abc = 2(abc − 2(ab + bc + ca) + 4(a + b + c) − 8).
Hence abc − 4(ab + bc + ca) + 8(a + b + c) − 16 = 0.

Geometrically, it's clear a > 3.  Now consider separately cases a = 4 to 9.

When a = 4:

−8(b + c) + 16 = 0
Therefore b + c = 2, contradicting a less than or equal to b

When a = 5:

bc − 12(b + c) + 24 = 0
(b − 12)(c − 12) = 120
Factorizing 120: b − 12 = (1, 2, 3, 4, 5, 6, 8, 10), c − 12 = (120, 60, 40, 30, 24, 20, 15, 12)
Therefore (b,c) = {(13,132), (14,72), (15,52), (16,42), (17,36), (18,32), (20,27), (22,24)}

When a = 6:

2bc − 16(b + c) + 32 = 0
bc − 8(b + c) + 16 = 0
(b − 8)(c − 8) = 48
Therefore (b,c) = {(9,56), (10,32), (11,24), (12,20), (14,16)}

When a = 7:

3bc − 20(b + c) + 40 = 0
9bc − 60(b + c) + 120 = 0
(3b − 20)(3c − 20) = 280
Therefore (b,c) = {(7,100), (8,30), (9,20), (10,16)}

When a = 8:

4bc − 24(b + c) + 48 = 0
bc − 6(b + c) + 12 = 0
(b − 6)(c − 6) = 24
Therefore (b,c) = {(8,18), (9,14), (10,12)}

When a = 9:

5bc − 28(b + c) + 56 = 0
25bc − 140(b + c) + 280 = 0
(5b − 28)(5c − 28) = 504
This has no solutions with b greater than or equal to 9.

Therefore there are 20 cuboids with number of internal cubes equal to number of external cubes; shown above; summarized below.

List of cuboids
abc# external cubes
5131324290
514722520
515521950
516421680
517361530
518321440
520271350
522241320
69561512
61032960
61124792
61220720
61416672
771002450
7830840
7920630
71016560
8818576
8914504
81012480

Further reading

  1. Online Encyclopedia of Integer Sequences: A115157.

Source: Inspired by the Peculiar Perimeter on mathschallenge.net

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