An A×B×C rectangular block has been divided into ABC unit cubes. Some of these cubes have faces on the surface of the block, while others are entirely interior.
Show that for any positive integer N there is a block with N times as many interior cubes as surface cubes.
Find the dimensions of such a block having the smallest volume.
clearvars
minvol=9999999999;
for tot=1:9999
for a=1:tot/3
for b=a:(tot-a)/2
c=tot-a-b;
if a>2 && b>2 && c>2
inner=(a-2)*(b-2)*(c-2);
outer=a*b*c-inner;
if inner/outer==round(inner/outer)
if a*b*c<=minvol
fprintf('%3d %4d %4d %4d %11d
',inner/outer, a, b, c, a*b*c)
minvol=a*b*c;
end
end
end
end
end
end
ratio a b c volume
1 8 10 12 960
Others found:
1 8 10 12 960
1 8 9 14 1008
1 7 10 16 1120
1 8 8 18 1152
1 6 14 16 1344
1 7 9 20 1260
1 6 12 20 1440
1 6 11 24 1584
1 7 8 30 1680
1 6 10 32 1920
2 14 16 18 4032
2 12 18 20 4320
2 13 15 22 4290
1 5 22 24 2640
1 5 20 27 2700
2 11 18 24 4752
2 12 15 26 4680
1 5 18 32 2880
2 10 22 24 5280
2 12 14 30 5040
2 10 20 27 5400
1 5 17 36 3060
2 10 18 32 5760
1 5 16 42 3360
2 9 26 28 6552
2 10 17 36 6120
2 9 22 35 6930
3 20 22 24 10560
3 20 20 27 10800
2 9 21 38 7182
2 10 16 42 6720
3 16 26 28 11648
3 18 20 32 11520
1 6 9 56 3024
2 9 20 42 7560
1 5 15 52 3900
3 15 26 32 12480
3 16 22 35 12320
3 17 20 36 12240
2 12 12 50 7200
3 16 21 38 12768
3 14 30 32 13440
3 17 19 40 12920
2 10 15 52 7800
2 8 34 36 9792
2 11 13 54 7722
3 16 20 42 13440
2 8 30 42 10080
3 14 24 44 14784
2 9 18 56 9072
4 26 28 30 21840
2 8 27 50 10800
3 13 28 44 16016
3 14 23 48 15456
3 15 20 52 15600
4 22 30 35 23100
2 8 26 54 11232
4 21 30 38 23940
3 12 38 40 18240
3 16 18 56 16128
4 20 34 36 24480
4 23 25 42 24150
1 5 14 72 5040
3 12 35 44 18480
4 20 30 42 25200
4 19 34 40 25840
4 22 25 46 25300
3 12 32 50 19200
2 9 17 70 10710
2 10 14 72 10080
4 18 38 40 27360
4 22 24 50 26400
4 18 35 44 27720
4 20 27 50 27000
2 8 24 66 12672
3 12 30 56 20160
4 18 32 50 28800
4 20 26 54 28080
3 12 29 60 20880
5 32 34 36 39168
3 16 17 70 19040
4 17 36 50 30600
4 18 30 56 30240
5 28 37 39 40404
5 30 32 42 40320
3 11 46 48 24288
3 12 28 65 21840
5 27 38 40 41040
3 14 20 72 20160
5 27 35 44 41580
4 18 29 60 31320
3 11 40 57 25080
4 16 42 50 33600
5 27 32 50 43200
4 20 24 66 31680
5 24 42 44 44352
5 28 30 52 43680
4 18 28 65 32760
5 26 32 54 44928
2 11 12 90 11880
5 23 42 48 46368
Edited on June 6, 2024, 10:46 am
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Posted by Charlie
on 2024-06-06 09:22:00 |
Part 2 only
