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 Make It Even Again (Posted on 2017-06-20)
Timothy and Urban are playing a dice game like they did before. As before the faces of the dice are colored red or blue but the dice could have any number of sides. Each die has at least 2 sides but the two dice do not necessarily the same number of faces. Both dice are fair.

The rules are the same: Timothy wins when the two top faces are the same color. Urban wins when the colors are different. Their chances are even with these dice.

Is it always the case that one of the dice has an equal number of red and blue faces?

 See The Solution Submitted by Brian Smith No Rating

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 computer findings | Comment 1 of 4
With up to 25-sided dice where the GCD of the number of blue faces and the total number of faces is 1 (or there are no blue faces), the cases where the probability of a match between the two dice is 1/2 are shown.  Each line shows blue faces/total faces for the first die, then for the second die, and a confirmation that the probability of a match is 1/2.  Each line shows that at least one die has 1/2 its faces blue.

This doesn't prove the case, but makes it likely. In practical terms dice with over 25 faces would be hard to make in a fair manner.

0   1/2   1/2
1/2   0   1/2
1/2   1/2   1/2
1/2   0   1/2
1/2   1/3   1/2
1/2   2/3   1/2
1/2   0   1/2
1/2   1/4   1/2
1/2   3/4   1/2
1/2   0   1/2
1/2   1/5   1/2
1/2   2/5   1/2
1/2   3/5   1/2
1/2   4/5   1/2
1/2   0   1/2
1/2   1/6   1/2
1/2   5/6   1/2
1/2   0   1/2
1/2   1/7   1/2
1/2   2/7   1/2
1/2   3/7   1/2
1/2   4/7   1/2
1/2   5/7   1/2
1/2   6/7   1/2
1/2   0   1/2
1/2   1/8   1/2
1/2   3/8   1/2
1/2   5/8   1/2
1/2   7/8   1/2
1/2   0   1/2
1/2   1/9   1/2
1/2   2/9   1/2
1/2   4/9   1/2
1/2   5/9   1/2
1/2   7/9   1/2
1/2   8/9   1/2
1/2   0   1/2
1/2   1/10   1/2
1/2   3/10   1/2
1/2   7/10   1/2
1/2   9/10   1/2
1/2   0   1/2
1/2   1/11   1/2
1/2   2/11   1/2
1/2   3/11   1/2
1/2   4/11   1/2
1/2   5/11   1/2
1/2   6/11   1/2
1/2   7/11   1/2
1/2   8/11   1/2
1/2   9/11   1/2
1/2   10/11   1/2
1/2   0   1/2
1/2   1/12   1/2
1/2   5/12   1/2
1/2   7/12   1/2
1/2   11/12   1/2
1/2   0   1/2
1/2   1/13   1/2
1/2   2/13   1/2
1/2   3/13   1/2
1/2   4/13   1/2
1/2   5/13   1/2
1/2   6/13   1/2
1/2   7/13   1/2
1/2   8/13   1/2
1/2   9/13   1/2
1/2   10/13   1/2
1/2   11/13   1/2
1/2   12/13   1/2
1/2   0   1/2
1/2   1/14   1/2
1/2   3/14   1/2
1/2   5/14   1/2
1/2   9/14   1/2
1/2   11/14   1/2
1/2   13/14   1/2
1/2   0   1/2
1/2   1/15   1/2
1/2   2/15   1/2
1/2   4/15   1/2
1/2   7/15   1/2
1/2   8/15   1/2
1/2   11/15   1/2
1/2   13/15   1/2
1/2   14/15   1/2
1/2   0   1/2
1/2   1/16   1/2
1/2   3/16   1/2
1/2   5/16   1/2
1/2   7/16   1/2
1/2   9/16   1/2
1/2   11/16   1/2
1/2   13/16   1/2
1/2   15/16   1/2
1/2   0   1/2
1/2   1/17   1/2
1/2   2/17   1/2
1/2   3/17   1/2
1/2   4/17   1/2
1/2   5/17   1/2
1/2   6/17   1/2
1/2   7/17   1/2
1/2   8/17   1/2
1/2   9/17   1/2
1/2   10/17   1/2
1/2   11/17   1/2
1/2   12/17   1/2
1/2   13/17   1/2
1/2   14/17   1/2
1/2   15/17   1/2
1/2   16/17   1/2
1/2   0   1/2
1/2   1/18   1/2
1/2   5/18   1/2
1/2   7/18   1/2
1/2   11/18   1/2
1/2   13/18   1/2
1/2   17/18   1/2
1/2   0   1/2
1/2   1/19   1/2
1/2   2/19   1/2
1/2   3/19   1/2
1/2   4/19   1/2
1/2   5/19   1/2
1/2   6/19   1/2
1/2   7/19   1/2
1/2   8/19   1/2
1/2   9/19   1/2
1/2   10/19   1/2
1/2   11/19   1/2
1/2   12/19   1/2
1/2   13/19   1/2
1/2   14/19   1/2
1/2   15/19   1/2
1/2   16/19   1/2
1/2   17/19   1/2
1/2   18/19   1/2
1/2   0   1/2
1/2   1/20   1/2
1/2   3/20   1/2
1/2   7/20   1/2
1/2   9/20   1/2
1/2   11/20   1/2
1/2   13/20   1/2
1/2   17/20   1/2
1/2   19/20   1/2
1/2   0   1/2
1/2   1/21   1/2
1/2   2/21   1/2
1/2   4/21   1/2
1/2   5/21   1/2
1/2   8/21   1/2
1/2   10/21   1/2
1/2   11/21   1/2
1/2   13/21   1/2
1/2   16/21   1/2
1/2   17/21   1/2
1/2   19/21   1/2
1/2   20/21   1/2
1/2   0   1/2
1/2   1/22   1/2
1/2   3/22   1/2
1/2   5/22   1/2
1/2   7/22   1/2
1/2   9/22   1/2
1/2   13/22   1/2
1/2   15/22   1/2
1/2   17/22   1/2
1/2   19/22   1/2
1/2   21/22   1/2
1/2   0   1/2
1/2   1/23   1/2
1/2   2/23   1/2
1/2   3/23   1/2
1/2   4/23   1/2
1/2   5/23   1/2
1/2   6/23   1/2
1/2   7/23   1/2
1/2   8/23   1/2
1/2   9/23   1/2
1/2   10/23   1/2
1/2   11/23   1/2
1/2   12/23   1/2
1/2   13/23   1/2
1/2   14/23   1/2
1/2   15/23   1/2
1/2   16/23   1/2
1/2   17/23   1/2
1/2   18/23   1/2
1/2   19/23   1/2
1/2   20/23   1/2
1/2   21/23   1/2
1/2   22/23   1/2
1/2   0   1/2
1/2   1/24   1/2
1/2   5/24   1/2
1/2   7/24   1/2
1/2   11/24   1/2
1/2   13/24   1/2
1/2   17/24   1/2
1/2   19/24   1/2
1/2   23/24   1/2
1/2   0   1/2
1/2   1/25   1/2
1/2   2/25   1/2
1/2   3/25   1/2
1/2   4/25   1/2
1/2   6/25   1/2
1/2   7/25   1/2
1/2   8/25   1/2
1/2   9/25   1/2
1/2   11/25   1/2
1/2   12/25   1/2
1/2   13/25   1/2
1/2   14/25   1/2
1/2   16/25   1/2
1/2   17/25   1/2
1/2   18/25   1/2
1/2   19/25   1/2
1/2   21/25   1/2
1/2   22/25   1/2
1/2   23/25   1/2
1/2   24/25   1/2
0   1/2   1/2
1/3   1/2   1/2
2/3   1/2   1/2
0   1/2   1/2
1/4   1/2   1/2
3/4   1/2   1/2
0   1/2   1/2
1/5   1/2   1/2
2/5   1/2   1/2
3/5   1/2   1/2
4/5   1/2   1/2
0   1/2   1/2
1/6   1/2   1/2
5/6   1/2   1/2
0   1/2   1/2
1/7   1/2   1/2
2/7   1/2   1/2
3/7   1/2   1/2
4/7   1/2   1/2
5/7   1/2   1/2
6/7   1/2   1/2
0   1/2   1/2
1/8   1/2   1/2
3/8   1/2   1/2
5/8   1/2   1/2
7/8   1/2   1/2
0   1/2   1/2
1/9   1/2   1/2
2/9   1/2   1/2
4/9   1/2   1/2
5/9   1/2   1/2
7/9   1/2   1/2
8/9   1/2   1/2
0   1/2   1/2
1/10   1/2   1/2
3/10   1/2   1/2
7/10   1/2   1/2
9/10   1/2   1/2
0   1/2   1/2
1/11   1/2   1/2
2/11   1/2   1/2
3/11   1/2   1/2
4/11   1/2   1/2
5/11   1/2   1/2
6/11   1/2   1/2
7/11   1/2   1/2
8/11   1/2   1/2
9/11   1/2   1/2
10/11   1/2   1/2
0   1/2   1/2
1/12   1/2   1/2
5/12   1/2   1/2
7/12   1/2   1/2
11/12   1/2   1/2
0   1/2   1/2
1/13   1/2   1/2
2/13   1/2   1/2
3/13   1/2   1/2
4/13   1/2   1/2
5/13   1/2   1/2
6/13   1/2   1/2
7/13   1/2   1/2
8/13   1/2   1/2
9/13   1/2   1/2
10/13   1/2   1/2
11/13   1/2   1/2
12/13   1/2   1/2
0   1/2   1/2
1/14   1/2   1/2
3/14   1/2   1/2
5/14   1/2   1/2
9/14   1/2   1/2
11/14   1/2   1/2
13/14   1/2   1/2
0   1/2   1/2
1/15   1/2   1/2
2/15   1/2   1/2
4/15   1/2   1/2
7/15   1/2   1/2
8/15   1/2   1/2
11/15   1/2   1/2
13/15   1/2   1/2
14/15   1/2   1/2
0   1/2   1/2
1/16   1/2   1/2
3/16   1/2   1/2
5/16   1/2   1/2
7/16   1/2   1/2
9/16   1/2   1/2
11/16   1/2   1/2
13/16   1/2   1/2
15/16   1/2   1/2
0   1/2   1/2
1/17   1/2   1/2
2/17   1/2   1/2
3/17   1/2   1/2
4/17   1/2   1/2
5/17   1/2   1/2
6/17   1/2   1/2
7/17   1/2   1/2
8/17   1/2   1/2
9/17   1/2   1/2
10/17   1/2   1/2
11/17   1/2   1/2
12/17   1/2   1/2
13/17   1/2   1/2
14/17   1/2   1/2
15/17   1/2   1/2
16/17   1/2   1/2
0   1/2   1/2
1/18   1/2   1/2
5/18   1/2   1/2
7/18   1/2   1/2
11/18   1/2   1/2
13/18   1/2   1/2
17/18   1/2   1/2
0   1/2   1/2
1/19   1/2   1/2
2/19   1/2   1/2
3/19   1/2   1/2
4/19   1/2   1/2
5/19   1/2   1/2
6/19   1/2   1/2
7/19   1/2   1/2
8/19   1/2   1/2
9/19   1/2   1/2
10/19   1/2   1/2
11/19   1/2   1/2
12/19   1/2   1/2
13/19   1/2   1/2
14/19   1/2   1/2
15/19   1/2   1/2
16/19   1/2   1/2
17/19   1/2   1/2
18/19   1/2   1/2
0   1/2   1/2
1/20   1/2   1/2
3/20   1/2   1/2
7/20   1/2   1/2
9/20   1/2   1/2
11/20   1/2   1/2
13/20   1/2   1/2
17/20   1/2   1/2
19/20   1/2   1/2
0   1/2   1/2
1/21   1/2   1/2
2/21   1/2   1/2
4/21   1/2   1/2
5/21   1/2   1/2
8/21   1/2   1/2
10/21   1/2   1/2
11/21   1/2   1/2
13/21   1/2   1/2
16/21   1/2   1/2
17/21   1/2   1/2
19/21   1/2   1/2
20/21   1/2   1/2
0   1/2   1/2
1/22   1/2   1/2
3/22   1/2   1/2
5/22   1/2   1/2
7/22   1/2   1/2
9/22   1/2   1/2
13/22   1/2   1/2
15/22   1/2   1/2
17/22   1/2   1/2
19/22   1/2   1/2
21/22   1/2   1/2
0   1/2   1/2
1/23   1/2   1/2
2/23   1/2   1/2
3/23   1/2   1/2
4/23   1/2   1/2
5/23   1/2   1/2
6/23   1/2   1/2
7/23   1/2   1/2
8/23   1/2   1/2
9/23   1/2   1/2
10/23   1/2   1/2
11/23   1/2   1/2
12/23   1/2   1/2
13/23   1/2   1/2
14/23   1/2   1/2
15/23   1/2   1/2
16/23   1/2   1/2
17/23   1/2   1/2
18/23   1/2   1/2
19/23   1/2   1/2
20/23   1/2   1/2
21/23   1/2   1/2
22/23   1/2   1/2
0   1/2   1/2
1/24   1/2   1/2
5/24   1/2   1/2
7/24   1/2   1/2
11/24   1/2   1/2
13/24   1/2   1/2
17/24   1/2   1/2
19/24   1/2   1/2
23/24   1/2   1/2
0   1/2   1/2
1/25   1/2   1/2
2/25   1/2   1/2
3/25   1/2   1/2
4/25   1/2   1/2
6/25   1/2   1/2
7/25   1/2   1/2
8/25   1/2   1/2
9/25   1/2   1/2
11/25   1/2   1/2
12/25   1/2   1/2
13/25   1/2   1/2
14/25   1/2   1/2
16/25   1/2   1/2
17/25   1/2   1/2
18/25   1/2   1/2
19/25   1/2   1/2
21/25   1/2   1/2
22/25   1/2   1/2
23/25   1/2   1/2
24/25   1/2   1/2

5   open "makeeven.txt" for output as #2
10   for D1=2 to 25
20      for N1=0 to D1
30         if gcd(N1,D1)=1 or N1=0 then
40           :for D2=2 to 25
50             :for N2=0 to D2
60               :if gcd(N2,D2)=1 or N2=0 then
70                 :P=N1//D1*N2//D2+(1-N1//D1)*(1-N2//D2)
80                 :if P=1//2 then print N1//D1,N2//D2,P
81                   :print #2,N1//D1,N2//D2,P
90                 :endif
100               :endif
110             :next
120           :next
130      next
140   next
150   close #2

 Posted by Charlie on 2017-06-20 15:18:19

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