IBM Research | Ponder This | December 2016 solutions

# Ponder This

## December 2016

<<November December January>>

All the solutions we present are built on splitting the N persons into groups and running the rope-pulling contest in subsets of these groups.
Here's an N=27 solution:
- ABCDEFGHI WXY Z ; JKLMNOPQ RSTUV
- ABCDEFGHI Z 0 ; JKLMNOPQ WXY
- ABCDEFGHI ; RSTUV WXY Z
- JKLMNOPQ Z ; RSTUV WXY 0

An N=30 solution:
- ABCD EFGHI JKLMNO ; PQRSTUV WXYZ0123
- ABCD PQRSTUV ; EFGHI JKLMNO
- JKLMNO PQRSTUV ; EFGHI WXYZ0123
- ABCD WXYZ0123 ; EFGHI PQRSTUV

To verify that the last solution works, you need to show that the integer solutions for the following equation set

```    ( 1  1  1 -1 -1 ) (x4)   (0)
( 1 -1 -1  1  0 ) (x5)   (0)
( 0 -1  1  1 -1 ) (x6) = (0)
( 1 -1  0 -1  1 ) (x7)   (0)
(x8)
```
are multiple of (4,5,6,7,8), which means that all the participants are from the same gender as required.

An N=114 solution for five rope-pulling contests can be achieved by
```     1 -1 -1  1 -1  1
-1  1  0  1 -1  0
1  1  0 -1 -1  1
1  0  1  1 -1 -1
1  1 -1  0  1 -1
```

with the only integer solution as follows:
9 12 20 22 25 26