Arrange the numbers 1...23 into three groups so that in any group the sum of two numbers from that group is not in that group.
combinatorics/sum.free.groups New
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Problem
Arrange the numbers 1...23 into three groups so that in any group the
sum of two numbers from that group is not in that group. Example: the
numbers 1, 2 and 3 may not be in the same group since 1+2=3. So the
numbers 1 to 6 may be arranged
1 // 2 // 3,4,5,6
Bonus: arrange the numbers 1 to 24.
Show solution
Running a small program found three solutions for the 23 puzzle and none for 24. The three solutions are: 1 2 4 8 11 16 22 3 5 6 7 19 21 23 9 10 12 13 14 15 17 18 20 1 2 4 8 11 17 22 3 5 6 7 19 21 23 9 10 12 13 14 15 16 18 20 1 2 4 8 11 22 3 5 6 7 19 21 23 9 10 12 13 14 15 16 17 18 20 Since any solution for `n' is also a solution for n-1, 23 is the biggest number for which this is possible. -- Amol V. Deshpande <amol@cse.iitb.ernet.in> [ Editor: the problem was posted by Ray Johnson. As the example 3,4,5,6 shows (3+3 = 6), the two numbers added must be different. ]