Pool Billiards Rack

Pool Billiards Rack

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

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Rack (Billiards) $71.7 High Quality Content by WIKIPEDIA articles A rack is the name given to a frame (usually aluminium, wood or plastic) used to organize billiard balls at the beginning of a game. Rack may also be used as a verb to describe the act of setting billiard balls in starting position in billiards games that make use of racks (usually, but not always, using a physical rack), as well as a noun to describe the balls in that starting position. The most common shape of a physical rack is that of a triangle, with the ball pattern of 54321. Racks are sometimes called simply triangles (most often by amateur shooters) based on the predominance of this form. Triangularshaped racks are used for eightball, straight pool, onepocket, bank pool, snooker and many other games. Although diamondshaped racks, with an intended pattern of 12321, are made for the game of nineball, the triangular rack is more often employed in nine ball as well. Author: Surhone, Lambert M./ Timpledon, Miriam T./ Marseken, Susan F. Binding Type: Paperback Number of Pages: 98 Publication Date: 2010/08/11 Language: English Dimensions: 6.00 x 9.02 x 0.23 inches

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