The letters, A to L, within this star represent intersections of unique pairings of its 6 lines, and α, β, γ, δ, ε and ζ are sums of intersections defined as:
α = A + D + G + K β = E + G + J + L γ = K + J + I + H
δ = L + I + F + B ε = H + F + C + A ζ = B + C + D + E
A α
/ \
ζ B---C---D---E β
\ / \ /
F G
/ \ / \
ε H---I---J---K γ
\ /
L δ
Assign values from 1 to 12 to each of the locations A to L such that
each sum is an element of an arithmetic progression with an arithmetic difference of two (2) but not necessarily as adjacent vertex values.
Secondly, attempt the same task but with a difference of four (4) as the outcome.
And for a tease... can you offer a solution if all such vertex sums are equal, ie, 26?
Note:
Discounting rotations and reflections, more than one possibility exists for each of the first two tasks.
(In reply to
computer solution; spoiler by Charlie)
The 72 solutions where each edge totals 26 is a small enough set to show them all:
2 | 2 | 2 | 2 | 3 | 3
| | | | |
3 12 5 6| 4 11 5 6| 4 12 7 3| 3 11 8 4| 2 10 5 9| 2 10 8 6
| | | | |
11 9 | 12 9 | 11 8 | 12 7 | 12 7 | 12 4
| | | | |
1 8 7 10| 1 7 8 10| 1 6 10 9| 1 6 10 9| 1 8 6 11| 1 5 9 11
| | | | |
4 | 3 | 5 | 5 | 4 | 7
3 | 4 | 4 | 4 | 5 | 5
| | | | |
4 12 8 2| 5 10 2 9| 5 12 3 6| 2 11 7 6| 2 12 3 9| 3 8 4 11
| | | | |
10 6 | 11 8 | 9 11 | 10 3 | 8 7 | 12 7
| | | | |
1 5 11 9| 1 7 6 12| 1 10 7 8| 1 5 8 12| 1 10 4 11| 1 9 6 10
| | | | |
7 | 3 | 2 | 9 | 6 | 2
5 | 5 | 5 | 5 | 5 | 5
| | | | |
3 11 4 8| 7 12 4 3| 7 9 6 4| 8 9 6 3| 4 9 7 6| 6 11 7 2
| | | | |
9 10 | 8 11 | 10 3 | 10 4 | 11 2 | 9 4
| | | | |
1 12 6 7| 1 9 10 6| 2 1 11 12| 2 1 12 11| 1 3 10 12| 1 3 12 10
| | | | |
2 | 2 | 8 | 7 | 8 | 8
5 | 5 | 5 | 5 | 5 | 6
| | | | |
4 6 9 7| 3 8 9 6| 6 7 9 4| 3 11 10 2| 2 6 11 7| 3 12 2 9
| | | | |
12 1 | 12 2 | 11 2 | 9 4 | 12 1 | 7 8
| | | | |
3 2 10 11| 1 4 11 10| 3 1 12 10| 1 6 12 7| 3 4 10 9| 1 11 4 10
| | | | |
8 | 7 | 8 | 8 | 8 | 5
6 | 6 | 6 | 6 | 6 | 6
| | | | |
2 11 3 10| 7 10 5 4| 8 10 5 3| 4 10 7 5| 4 12 8 2| 4 7 10 5
| | | | |
8 5 | 9 3 | 9 4 | 8 1 | 7 3 | 11 1
| | | | |
1 9 4 12| 1 2 11 12| 1 2 12 11| 2 3 9 12| 1 5 11 9| 2 3 12 9
| | | | |
7 | 8 | 7 | 11 | 10 | 8
6 | 6 | 7 | 7 | 7 | 7
| | | | |
5 7 11 3| 2 8 12 4| 3 10 2 11| 4 8 2 12|12 4 2 8| 9 8 3 6
| | | | |
9 1 | 9 1 | 8 5 | 10 6 | 10 6 | 10 4
| | | | |
4 2 12 8| 3 5 11 7| 1 9 4 12| 1 9 5 11| 5 1 9 11| 1 2 11 12
| | | | |
10 | 10 | 6 | 3 | 3 | 5
7 | 7 | 7 | 7 | 7 | 7
| | | | |
0 8 3 5| 2 12 3 9| 8 6 3 9| 2 8 4 12| 9 1 5 11| 8 3 6 9
| | | | |
9 4 | 6 5 | 12 5 | 10 6 | 12 4 | 12 2
| | | | |
2 1 11 12| 1 10 4 11| 1 4 10 11| 1 11 5 9| 6 2 8 10| 4 1 10 11
| | | | |
6 | 8 | 2 | 3 | 3 | 5
7 | 7 | 7 | 8 | 8 | 8
| | | | |
0 2 6 8| 3 11 8 4| 5 9 10 2|11 3 2 10|11 6 2 7| 3 12 2 9
| | | | |
12 4 | 6 1 | 6 1 | 9 4 | 9 4 | 5 6
| | | | |
5 1 11 9| 2 5 9 10| 4 3 11 8| 6 1 7 12| 3 1 10 12| 1 11 4 10
| | | | |
3 | 12 | 12 | 5 | 5 | 7
8 | 8 | 8 | 8 | 8 | 8
| | | | |
3 11 2 10| 2 10 3 11| 7 6 4 9| 9 1 4 12| 9 7 4 6|10 7 4 5
| | | | |
6 7 | 7 6 | 11 2 | 10 3 | 10 3 | 9 3
| | | | |
1 12 4 9| 1 12 4 9| 1 3 10 12| 7 2 6 11| 1 2 12 11| 2 1 12 11
| | | | |
5 | 5 | 5 | 5 | 5 | 6
8 | 8 | 8 | 8 | 8 | 8
| | | | |
2 9 5 10| 7 4 5 10| 9 4 6 7| 6 2 7 11| 6 4 7 9| 4 12 7 3
| | | | |
6 1 | 11 1 | 11 2 | 12 1 | 12 1 | 5 2
| | | | |
3 7 4 12| 3 2 9 12| 3 1 12 10| 4 3 9 10| 2 3 11 10| 1 6 10 9
| | | | |
11 | 6 | 5 | 5 | 5 | 11
9 | 9 | 9 | 9 | 9 | 10
| | | | |
7 6 2 11| 4 11 3 8| 6 5 3 12| 2 7 5 12| 3 12 5 6| 4 8 2 12
| | | | |
10 3 | 5 2 | 11 4 | 6 1 | 4 2 | 7 3
| | | | |
1 5 8 12| 1 7 6 12| 1 7 8 10| 4 8 3 11| 1 8 7 10| 1 9 5 11
| | | | |
4 | 10 | 2 | 10 | 11 | 6
10 | 10 | 10 | 10 | 11 | 11
| | | | |
2 4 2 8| 4 8 3 11| 7 5 3 11| 8 2 4 12| 7 8 2 9| 9 5 2 10
| | | | |
7 3 | 6 1 | 9 1 | 9 1 | 4 1 | 6 1
| | | | |
5 1 9 11| 2 7 5 12| 2 4 8 12| 5 3 7 11| 3 5 6 12| 4 3 7 12
| | | | |
6 | 9 | 6 | 6 | 10 | 8
|
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Posted by Charlie
on 2008-06-02 00:06:57 |
re: computer solution; spoiler
