**Source:**Shared by Alok Mittal (Cannan Partners)

**Problem:**

Update (23 Oct 2014)

**Solution:**Posted in comments by Anti, Taz, Javier, Shubham Gupta, Leela. Detailed solution and much more advanced problems in the document http://library.msri.org/books/Book42/files/golomb.pdf

9 per face, 3 per edge, 1 per corner, divide by 2 for symmetry: (9*6 + 3*12 + 8)/2 --> 49

ReplyDelete24 (8*3 Imagine 3 2D plates stacked horizontally)

ReplyDelete24 (8*3 Imagine 3 2D plates stacked vertically)

4 (Diagonals of the cube)

Total of 52 winning lines

9 different cross-sections of dimension 1x3x3, each having 8 winning lines, plus 4 body diagonals. Hence, 9*8+4=76.

ReplyDelete3 layers each of 3 rt angle faces. Each layer is a 3*3 tic tac toe

ReplyDelete# of winning lines = 3*3*8 = 72

Total- 43

ReplyDelete(24 - When 3 2D tic-tac planes are vertically stacked)

(15 - When 3 2D tic-tac planes are horizontally stacked minus number of winning lines already covered in previous case)

(4 - Diagonal winning lines)

ReplyDeleteFor a simple 2 Dimensional Grid: 3 in x-direction + 3 along y +2 diagonals = 8 solutions

Extending same for 3 Dimensional Grid:

3*3 along x + 3*3along y +3*3 along z + 2*3 diagonals along xy + 2*3 diagonals along xz + 2*3 yz diagonals + 4 other diagonals = 49

I found this: http://library.msri.org/books/Book42/files/golomb.pdf

ReplyDeleteIs it 82 ?

ReplyDelete72 + 4 + 6.

It won't be 82. Lots of repetition in 82.

DeleteAnswer should be 49.

ReplyDeleteTop and bottom layer -

_______

|7|3|7|

|3|1|3|

|7|3|7|

_______

mid layer -

_______

|3|1|3|

|1|0|1|

|3|1|3|

_______

each matrix component tells about number of winning lines start/end at that cell.

sum = 98

ans = 98/2 = 49 (same line counted twice, from each cell it can start or end)

Winning lines will always be in direction along with x,y,z axis or 45 aligned from them .

Answer should be 49.

ReplyDeleteTop and bottom layer -

_______

|7|3|7|

|3|1|3|

|7|3|7|

_______

mid layer -

_______

|3|1|3|

|1|0|1|

|3|1|3|

_______

each matrix component tells about number of winning lines start/end at that cell.

sum = 98

ans = 98/2 = 49 (same line counted twice, from each cell it can start or end)

Winning lines will always be in direction along with x,y,z axis or 45 aligned from them .

76.

ReplyDelete8 winning lines * 9 cross sections + 4 body diagonals

48 (8*6 faces) + 13 (26 points touching center and 2 will form a line with center) = 61 total

ReplyDelete49

ReplyDeleteThis is a very interesting question i must say.

ReplyDeleteOne may compute the answer as 49 or 76 by two different approaches. Which one is correct, becomes a new question

I first computed 76 as the obvious answer: (8 winning lines per face) x (3 faces per direction) x (3 directions) + (4 Body diagonals) = 76

Then I looked at the approach of 49 as answer very easily illustrated in answer by Shubham Gupta which seems to be correct too.

So again 49 vs 76. Which is it. Take your pick!

Now think a bit carefully.

When one counts 76 winning lines, one considers many repetitions.

Focus on the front-top-left cube piece. When we count 76 winning lines, we count it three times.

Think it in the same way as a rubic cube has 27 pieces ( hence our tic-tac-toe has only 27 possible places to make a move) whereas it has 9*6 coloured stickers.

So when we consider 76 lines we are considering the stickers which is incorrect.