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challenge billiards problem (algebra 2)?
Fifteen pool balls are numbered 1 to 15.
Arrange them in a triangular form (as if they were in a rack), so that the three balls in the middle sum to 39 and the 5 balls on each side sum to 39.
also show me how you got the answer too plz (show work)
Sum of all of them = 16 * 15 / 2 = 120
The sum of the 3 in the middle + the 3 edges = 4 * 39 = 156
with the 3 in the corners counted twice.
So their sum = 36.
So we want 3 that sum to 39 for the middle
and 3 that sum to 36 for the corners, and the possibilities are:
36 = 7 14 15
36 = 8 13 15
36 = 9 12 15
36 = 9 13 14
36 = 10 11 15
36 = 10 12 14
36 = 11 12 13
39 = 10 14 15
39 = 11 13 15
39 = 12 13 14
We need one of each, with none in common, which are limited to these:
Corners . . . . . . . Middle
36 = 10 11 15 + 39 = 12 13 14
36 = 10 12 14 + 39 = 11 13 15
36 = 11 12 13 + 39 = 10 14 15
Corners: .... Remaining edge sums
10 11 15 ... 39 - 21 = 18, 39 - 26 = 13, 39 - 25 = 14
10 12 14 ... 17, 13, 15
11 12 13 ... 16, 14, 15
I'm inclined to try the last one first, since it is the most balanced.
We have left 1-9 to make 14 15 16
1 5 9 = 15
2 4 8 = 14
3 6 7 = 16
And we're done: . . . . . Can swap these around:
13 .1 . 5 . 9 . 11
.. 2 10 15 ..3 . . . . . . . . .7+3 with 8+2 + 9+1
... 4. 14 . 6. . . . . . . . . .. . 4 + 2 with 5 + 1
.. ..8 ... 7
. . . .12
The other corners-middle combinations aren't any harder:
10 9 8 1 11
.6 12 13 7 . . . . . . . . . . . 8+1 with 6+3 with 7+2
. 5 14 4 . . . . . . . . . . . . . . . .7+4 with 6+5
. .3. 2
. . 15
and
12 9 7 1 10
.6 11 13 8 . . . . . . . . . . . . . 8+2 with 6+4 with 9+1
. 4 15 5 . . . . . . . . . . . . . . .. . .6+4 with 8+2
. . 3 2 . . . . . . . . . . . .. . . . . . ..4+3 with 5+2
. . 14
Corey Dueul's Challenge on Running A Rack of Balls Pool & Billiard