6*3 & 4*5; 6*4 & 3*5; 5*6 & 4*3 .
The 3 couples above represent six domino tiles chosen in a way such that multiplying the 4 numbers represented by pips of each couple one gets the same product , in our case 360.
If we consider multiplying the digits on both sides of "&" we get 12, 12, and 24, summing up to 48.
Just by changing the order within 1st and 2nd couples this sum can be increased to
84 ( 3*6 & 5*4; 4*6 & 5*3; 5*6 & 4*3).
Problem:
Out of a standard 28-tiles domino set choose 2k tiles selected so that they can be placed in k
rows, the product of pips in each row being the same.
For each row choose the highest numbers on each tile and multiply them.
Evaluate the sum of all products.
You are requested to provide the selection with the maximal sum.
It seems the set where all four pips multiply to 120 has the highest sought total, 149, obtained by perusing the below output:
In order of the product of all four numbers for the pairs of dominoes:
product product of
of high all four
domino 1 domino 2 pips pips
1 1 1 2 2 2
1 1 1 3 3 3
1 1 2 2 2 4
1 1 1 4 4 4
1 1 1 5 5 5
1 1 1 6 6 6
1 1 2 3 3 6
1 2 1 3 6 6
1 1 2 4 4 8
1 2 1 4 8 8
1 2 2 2 4 8
1 1 3 3 3 9
1 2 1 5 10 10
1 1 2 5 5 10
1 3 2 2 6 12
1 3 1 4 12 12
1 1 2 6 6 12
1 1 3 4 4 12
1 2 1 6 12 12
1 2 2 3 6 12
1 3 1 5 15 15
1 1 3 5 5 15
1 1 4 4 4 16
1 4 2 2 8 16
1 2 2 4 8 16
1 3 2 3 9 18
1 2 3 3 6 18
1 3 1 6 18 18
1 1 3 6 6 18
1 2 2 5 10 20
1 4 1 5 20 20
1 1 4 5 5 20
2 2 1 5 10 20
2 2 2 3 6 24
1 3 2 4 12 24
1 2 3 4 8 24
1 4 1 6 24 24
1 2 2 6 12 24
1 1 4 6 6 24
2 2 1 6 12 24
1 4 2 3 12 24
1 1 5 5 5 25
1 3 3 3 9 27
1 5 2 3 15 30
1 5 1 6 30 30
1 3 2 5 15 30
1 2 3 5 10 30
1 1 5 6 6 30
1 4 2 4 16 32
2 2 2 4 8 32
1 2 4 4 8 32
1 4 3 3 12 36
1 3 3 4 12 36
1 3 2 6 18 36
2 2 3 3 6 36
1 6 2 3 18 36
1 2 3 6 12 36
1 1 6 6 6 36
1 2 4 5 10 40
1 4 2 5 20 40
1 5 2 4 20 40
2 2 2 5 10 40
1 5 3 3 15 45
1 3 3 5 15 45
1 4 2 6 24 48
1 4 3 4 16 48
1 2 4 6 12 48
1 3 4 4 12 48
1 6 2 4 24 48
2 2 2 6 12 48
2 2 3 4 8 48
2 3 2 4 12 48
1 5 2 5 25 50
1 2 5 5 10 50
1 3 3 6 18 54
2 3 3 3 9 54
1 6 3 3 18 54
1 6 2 5 30 60
1 2 5 6 12 60
1 3 4 5 15 60
2 3 2 5 15 60
1 4 3 5 20 60
2 2 3 5 10 60
1 5 3 4 20 60
1 5 2 6 30 60
2 2 4 4 8 64
1 4 4 4 16 64
2 3 3 4 12 72
2 2 3 6 12 72
1 4 3 6 24 72
1 6 3 4 24 72
1 6 2 6 36 72
2 3 2 6 18 72
1 3 4 6 18 72
1 2 6 6 12 72
2 4 3 3 12 72
1 5 3 5 25 75
1 3 5 5 15 75
2 2 4 5 10 80
1 5 4 4 20 80
1 4 4 5 20 80
2 4 2 5 20 80
3 3 2 5 15 90
2 3 3 5 15 90
1 3 5 6 18 90
1 5 3 6 30 90
1 6 3 5 30 90
2 2 4 6 12 96
1 4 4 6 24 96
1 6 4 4 24 96
2 3 4 4 12 96
2 4 3 4 16 96
2 4 2 6 24 96
1 4 5 5 20 100
2 2 5 5 10 100
1 5 4 5 25 100
3 3 2 6 18 108
3 3 3 4 12 108
1 6 3 6 36 108
2 3 3 6 18 108
1 3 6 6 18 108
2 4 3 5 20 120
1 4 5 6 24 120
2 2 5 6 12 120 (not counted: reuses 5,6)
2 5 3 4 20 120 (not counted: 2,5 is better used below)
1 6 4 5 30 120
2 5 2 6 30 120
2 3 4 5 15 120
1 5 4 6 30 120 149
1 5 5 5 25 125
2 4 4 4 16 128
3 3 3 5 15 135
3 3 4 4 12 144
2 3 4 6 18 144
2 2 6 6 12 144
1 4 6 6 24 144
2 6 3 4 24 144
2 4 3 6 24 144
1 6 4 6 36 144
2 3 5 5 15 150
1 6 5 5 30 150
1 5 5 6 30 150
2 5 3 5 25 150
2 4 4 5 20 160
2 5 4 4 20 160
3 3 3 6 18 162
2 5 3 6 30 180
2 3 5 6 18 180
1 6 5 6 36 180
1 5 6 6 30 180
3 3 4 5 15 180
2 6 3 5 30 180
3 4 3 5 20 180 141
2 6 4 4 24 192
3 4 4 4 16 192
2 4 4 6 24 192
2 5 4 5 25 200
2 4 5 5 20 200
2 3 6 6 18 216
3 4 3 6 24 216
3 3 4 6 18 216
2 6 3 6 36 216
1 6 6 6 36 216
3 3 5 5 15 225
2 4 5 6 24 240
3 4 4 5 20 240
2 5 4 6 30 240
3 5 4 4 20 240
2 6 4 5 30 240
2 5 5 5 25 250
3 5 3 6 30 270
3 3 5 6 18 270
3 4 4 6 24 288
2 4 6 6 24 288
2 6 4 6 36 288
4 4 3 6 24 288
3 5 4 5 25 300
2 6 5 5 30 300
3 4 5 5 20 300
2 5 5 6 30 300
4 4 4 5 20 320
3 3 6 6 18 324
2 5 6 6 30 360
2 6 5 6 36 360
3 5 4 6 30 360
3 6 4 5 30 360
3 4 5 6 24 360
3 5 5 5 25 375
4 4 4 6 24 384
4 4 5 5 20 400
3 4 6 6 24 432
2 6 6 6 36 432
3 6 4 6 36 432
3 5 5 6 30 450
3 6 5 5 30 450
4 5 4 6 30 480
4 4 5 6 24 480
4 5 5 5 25 500
3 6 5 6 36 540
3 5 6 6 30 540
4 4 6 6 24 576
4 5 5 6 30 600
4 6 5 5 30 600
3 6 6 6 36 648
4 6 5 6 36 720
4 5 6 6 30 720
5 5 5 6 30 750
4 6 6 6 36 864
5 5 6 6 30 900
5 6 6 6 36 1080
The only set not considered was the zero set, where 0,1 through 0,6 could be paired with 6,1 through 6,6, but that set only adds to 21*6 = 126.
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Posted by Charlie
on 2013-07-29 17:36:08 |
computer assisted solution
