This is the net map of the outer surfaces of the 20 edge cubes that represents
a 3 x 3 x 3 'parent' cube (like in a Rubik cube).
Assign unique values from 1 to 20 to the letters A-T such that the sum of each pair of diametrically opposite cubes is to be the same as all others while the sum of each set of edge cubes may not differ from that of any other set by more than one.
Eg, diagonals:
A + Q = N + E = C + O ...(etc)
and edge cubes: A + B + C = C + E + H = F + G + H ...etc,
or (A + B + C) ±1 = C + E + H
= F + G + H ....(etc)
Note: The problem's development and my solution used a spreadsheet; as such a well-constructed sheet could enable a solution. Although this problem may lend itself to a programmed solution I would appreciate seeing attempts of a more manual basis within the first 24-48 hrs.
The layouts for 44 solutions follow. The sequences of 31's and 32's are for visual confirmation that no set of edge totals differs from any other by more than one, as the program only checked that opposite edges don't differ by more than 1. It turns out that this is sufficient, as I don't see any numbers other than 31 and 32.
The program checked for uniqueness of the solution by assuring that A is the lowest numbered corner and around A, the sequence from low to high was B<D<I. Other solutions are just a rotation/reflection of one of these.
1
1 11 19
13 0 5
18 6 7
1 13 18 18 6 7 7 5 19 19 11 1
17 0 12 12 0 4 4 0 9 9 0 17
14 16 2 2 10 20 20 8 3 3 15 14
2 10 20
16 0 8
14 15 3
31 32 32 31 31 32 31 32 32 31 32 31
2
1 11 19
14 0 4
16 6 9
1 14 16 16 6 9 9 4 19 19 11 1
18 0 13 13 0 3 3 0 8 8 0 18
12 17 2 2 10 20 20 7 5 5 15 12
2 10 20
17 0 7
12 15 5
31 32 31 32 32 31 31 32 31 32 31 32
3
1 11 19
14 0 5
17 6 8
1 14 17 17 6 8 8 5 19 19 11 1
18 0 12 12 0 3 3 0 9 9 0 18
13 16 2 2 10 20 20 7 4 4 15 13
2 10 20
16 0 7
13 15 4
31 32 32 31 32 31 31 32 32 31 31 32
4
1 11 19
15 0 4
16 7 9
1 15 16 16 7 9 9 4 19 19 11 1
18 0 13 13 0 3 3 0 8 8 0 18
12 17 2 2 10 20 20 6 5 5 14 12
2 10 20
17 0 6
12 14 5
31 32 32 31 32 31 32 31 31 32 31 32
5
1 11 19
16 0 4
14 9 8
1 16 14 14 9 8 8 4 19 19 11 1
18 0 15 15 0 3 3 0 6 6 0 18
13 17 2 2 10 20 20 5 7 7 12 13
2 10 20
17 0 5
13 12 7
31 32 31 32 31 32 31 32 32 31 31 32
6
1 11 19
16 0 4
14 8 9
1 16 14 14 8 9 9 4 19 19 11 1
18 0 15 15 0 3 3 0 6 6 0 18
12 17 2 2 10 20 20 5 7 7 13 12
2 10 20
17 0 5
12 13 7
31 32 31 32 32 31 31 32 31 32 31 32
7
1 11 19
16 0 4
15 9 8
1 16 15 15 9 8 8 4 19 19 11 1
18 0 14 14 0 3 3 0 7 7 0 18
13 17 2 2 10 20 20 5 6 6 12 13
2 10 20
17 0 5
13 12 6
31 32 32 31 31 32 32 31 32 31 31 32
8
1 11 19
16 0 4
15 8 9
1 16 15 15 8 9 9 4 19 19 11 1
18 0 14 14 0 3 3 0 7 7 0 18
12 17 2 2 10 20 20 5 6 6 13 12
2 10 20
17 0 5
12 13 6
31 32 32 31 32 31 32 31 31 32 31 32
9
1 11 19
17 0 5
14 9 8
1 17 14 14 9 8 8 5 19 19 11 1
18 0 15 15 0 3 3 0 6 6 0 18
13 16 2 2 10 20 20 4 7 7 12 13
2 10 20
16 0 4
13 12 7
31 32 32 31 32 31 31 32 32 31 31 32
10
1 12 18
14 0 4
16 6 10
1 14 16 16 6 10 10 4 18 18 12 1
19 0 13 13 0 2 2 0 8 8 0 19
11 17 3 3 9 20 20 7 5 5 15 11
3 9 20
17 0 7
11 15 5
31 32 31 32 32 31 32 31 31 32 32 31
11
1 12 18
16 0 4
14 8 10
1 16 14 14 8 10 10 4 18 18 12 1
19 0 15 15 0 2 2 0 6 6 0 19
11 17 3 3 9 20 20 5 7 7 13 11
3 9 20
17 0 5
11 13 7
31 32 31 32 32 31 32 31 31 32 32 31
12
1 12 18
16 0 4
15 7 10
1 16 15 15 7 10 10 4 18 18 12 1
19 0 13 13 0 2 2 0 8 8 0 19
11 17 3 3 9 20 20 5 6 6 14 11
3 9 20
17 0 5
11 14 6
31 32 32 31 32 31 32 31 31 32 31 32
13
1 12 19
13 0 5
18 6 7
1 13 18 18 6 7 7 5 19 19 12 1
17 0 11 11 0 4 4 0 10 10 0 17
14 16 2 2 9 20 20 8 3 3 15 14
2 9 20
16 0 8
14 15 3
32 31 32 31 31 32 31 32 32 31 31 32
14
1 12 19
16 0 4
14 10 8
1 16 14 14 10 8 8 4 19 19 12 1
18 0 15 15 0 3 3 0 6 6 0 18
13 17 2 2 9 20 20 5 7 7 11 13
2 9 20
17 0 5
13 11 7
32 31 31 32 31 32 32 31 32 31 31 32
15
1 12 19
17 0 5
14 10 8
1 17 14 14 10 8 8 5 19 19 12 1
18 0 15 15 0 3 3 0 6 6 0 18
13 16 2 2 9 20 20 4 7 7 11 13
2 9 20
16 0 4
13 11 7
32 31 32 31 32 31 32 31 32 31 31 32
16
1 13 18
14 0 4
16 6 10
1 14 16 16 6 10 10 4 18 18 13 1
19 0 12 12 0 2 2 0 9 9 0 19
11 17 3 3 8 20 20 7 5 5 15 11
3 8 20
17 0 7
11 15 5
32 31 31 32 32 31 32 31 31 32 31 32
17
1 13 18
14 0 5
17 6 9
1 14 17 17 6 9 9 5 18 18 13 1
19 0 11 11 0 2 2 0 10 10 0 19
12 16 3 3 8 20 20 7 4 4 15 12
3 8 20
16 0 7
12 15 4
32 31 32 31 32 31 32 31 32 31 31 32
18
1 14 17
18 0 5
13 10 9
1 18 13 13 10 9 9 5 17 17 14 1
19 0 15 15 0 2 2 0 6 6 0 19
12 16 4 4 7 20 20 3 8 8 11 12
4 7 20
16 0 3
12 11 8
32 31 32 31 31 32 32 31 32 31 32 31
19
1 14 17
18 0 5
13 9 10
1 18 13 13 9 10 10 5 17 17 14 1
19 0 15 15 0 2 2 0 6 6 0 19
11 16 4 4 7 20 20 3 8 8 12 11
4 7 20
16 0 3
11 12 8
32 31 32 31 32 31 32 31 31 32 32 31
20
2 12 17
14 0 3
15 5 11
2 14 15 15 5 11 11 3 17 17 12 2
20 0 13 13 0 1 1 0 8 8 0 20
10 18 4 4 9 19 19 7 6 6 16 10
4 9 19
18 0 7
10 16 6
31 32 31 32 31 32 31 32 32 31 32 31
21
2 12 17
16 0 3
13 7 11
2 16 13 13 7 11 11 3 17 17 12 2
20 0 15 15 0 1 1 0 6 6 0 20
10 18 4 4 9 19 19 5 8 8 14 10
4 9 19
18 0 5
10 14 8
31 32 31 32 31 32 31 32 32 31 32 31
22
2 12 17
16 0 3
14 6 11
2 16 14 14 6 11 11 3 17 17 12 2
20 0 13 13 0 1 1 0 8 8 0 20
10 18 4 4 9 19 19 5 7 7 15 10
4 9 19
18 0 5
10 15 7
31 32 32 31 31 32 31 32 32 31 31 32
23
2 13 16
14 0 3
15 4 12
2 14 15 15 4 12 12 3 16 16 13 2
20 0 11 11 0 1 1 0 10 10 0 20
9 18 5 5 8 19 19 7 6 6 17 9
5 8 19
18 0 7
9 17 6
31 32 31 32 31 32 31 32 31 32 31 32
24
2 13 17
14 0 3
15 5 11
2 14 15 15 5 11 11 3 17 17 13 2
20 0 12 12 0 1 1 0 9 9 0 20
10 18 4 4 8 19 19 7 6 6 16 10
4 8 19
18 0 7
10 16 6
32 31 31 32 31 32 31 32 32 31 31 32
25
2 14 16
18 0 4
11 8 12
2 18 11 11 8 12 12 4 16 16 14 2
20 0 15 15 0 1 1 0 6 6 0 20
9 17 5 5 7 19 19 3 10 10 13 9
5 7 19
17 0 3
9 13 10
32 31 31 32 32 31 31 32 31 32 31 32
26
2 14 16
18 0 4
12 8 11
2 18 12 12 8 11 11 4 16 16 14 2
20 0 15 15 0 1 1 0 6 6 0 20
10 17 5 5 7 19 19 3 9 9 13 10
5 7 19
17 0 3
10 13 9
32 31 32 31 31 32 31 32 32 31 32 31
27
3 11 17
14 0 2
15 5 12
3 14 15 15 5 12 12 2 17 17 11 3
20 0 13 13 0 1 1 0 8 8 0 20
9 19 4 4 10 18 18 7 6 6 16 9
4 10 18
19 0 7
9 16 6
31 32 32 31 31 32 32 31 32 31 32 31
28
3 11 17
15 0 2
14 5 12
3 15 14 14 5 12 12 2 17 17 11 3
20 0 13 13 0 1 1 0 8 8 0 20
9 19 4 4 10 18 18 6 7 7 16 9
4 10 18
19 0 6
9 16 7
31 32 32 31 31 32 31 32 32 31 31 32
29
3 11 17
16 0 2
12 7 13
3 16 12 12 7 13 13 2 17 17 11 3
20 0 15 15 0 1 1 0 6 6 0 20
8 19 4 4 10 18 18 5 9 9 14 8
4 10 18
19 0 5
8 14 9
31 32 31 32 32 31 32 31 31 32 31 32
30
3 11 17
16 0 2
13 6 12
3 16 13 13 6 12 12 2 17 17 11 3
20 0 14 14 0 1 1 0 7 7 0 20
9 19 4 4 10 18 18 5 8 8 15 9
4 10 18
19 0 5
9 15 8
31 32 32 31 31 32 31 32 32 31 31 32
31
3 11 17
16 0 2
13 7 12
3 16 13 13 7 12 12 2 17 17 11 3
20 0 15 15 0 1 1 0 6 6 0 20
9 19 4 4 10 18 18 5 8 8 14 9
4 10 18
19 0 5
9 14 8
31 32 32 31 31 32 32 31 32 31 32 31
32
3 12 16
14 0 2
15 4 13
3 14 15 15 4 13 13 2 16 16 12 3
20 0 11 11 0 1 1 0 10 10 0 20
8 19 5 5 9 18 18 7 6 6 17 8
5 9 18
19 0 7
8 17 6
31 32 32 31 31 32 32 31 31 32 31 32
33
3 12 16
17 0 2
11 7 13
3 17 11 11 7 13 13 2 16 16 12 3
20 0 15 15 0 1 1 0 6 6 0 20
8 19 5 5 9 18 18 4 10 10 14 8
5 9 18
19 0 4
8 14 10
31 32 31 32 31 32 31 32 31 32 31 32
34
3 14 15
19 0 4
9 10 13
3 19 9 9 10 13 13 4 15 15 14 3
20 0 16 16 0 1 1 0 5 5 0 20
8 17 6 6 7 18 18 2 12 12 11 8
6 7 18
17 0 2
8 11 12
32 31 31 32 32 31 32 31 31 32 31 32
35
3 14 15
19 0 4
10 9 13
3 19 10 10 9 13 13 4 15 15 14 3
20 0 16 16 0 1 1 0 5 5 0 20
8 17 6 6 7 18 18 2 11 11 12 8
6 7 18
17 0 2
8 12 11
32 31 32 31 32 31 32 31 31 32 32 31
36
3 15 14
19 0 4
9 10 13
3 19 9 9 10 13 13 4 14 14 15 3
20 0 16 16 0 1 1 0 5 5 0 20
8 17 7 7 6 18 18 2 12 12 11 8
7 6 18
17 0 2
8 11 12
32 31 31 32 31 32 32 31 31 32 32 31
37
4 11 16
13 0 2
15 3 14
4 13 15 15 3 14 14 2 16 16 11 4
20 0 12 12 0 1 1 0 9 9 0 20
7 19 5 5 10 17 17 8 6 6 18 7
5 10 17
19 0 8
7 18 6
31 32 32 31 32 31 32 31 31 32 32 31
38
4 12 15
18 0 2
10 8 14
4 18 10 10 8 14 14 2 15 15 12 4
20 0 16 16 0 1 1 0 5 5 0 20
7 19 6 6 9 17 17 3 11 11 13 7
6 9 17
19 0 3
7 13 11
31 32 32 31 31 32 32 31 31 32 32 31
39
4 12 16
13 0 2
15 3 14
4 13 15 15 3 14 14 2 16 16 12 4
20 0 11 11 0 1 1 0 10 10 0 20
7 19 5 5 9 17 17 8 6 6 18 7
5 9 17
19 0 8
7 18 6
32 31 32 31 32 31 32 31 31 32 31 32
40
5 11 15
18 0 2
8 9 14
5 18 8 8 9 14 14 2 15 15 11 5
20 0 17 17 0 1 1 0 4 4 0 20
7 19 6 6 10 16 16 3 13 13 12 7
6 10 16
19 0 3
7 12 13
31 32 31 32 31 32 31 32 32 31 31 32
41
5 11 15
18 0 2
9 8 14
5 18 9 9 8 14 14 2 15 15 11 5
20 0 17 17 0 1 1 0 4 4 0 20
7 19 6 6 10 16 16 3 12 12 13 7
6 10 16
19 0 3
7 13 12
31 32 32 31 31 32 31 32 32 31 32 31
42
5 11 15
19 0 3
8 9 14
5 19 8 8 9 14 14 3 15 15 11 5
20 0 17 17 0 1 1 0 4 4 0 20
7 18 6 6 10 16 16 2 13 13 12 7
6 10 16
18 0 2
7 12 13
31 32 32 31 32 31 31 32 32 31 31 32
43
5 12 15
18 0 2
8 10 14
5 18 8 8 10 14 14 2 15 15 12 5
20 0 17 17 0 1 1 0 4 4 0 20
7 19 6 6 9 16 16 3 13 13 11 7
6 9 16
19 0 3
7 11 13
32 31 31 32 31 32 32 31 32 31 31 32
44
5 12 15
19 0 3
8 10 14
5 19 8 8 10 14 14 3 15 15 12 5
20 0 17 17 0 1 1 0 4 4 0 20
7 18 6 6 9 16 16 2 13 13 11 7
6 9 16
18 0 2
7 11 13
32 31 32 31 32 31 32 31 32 31 31 32
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Posted by Charlie
on 2009-05-01 11:59:18 |
computer solutions
