# Miscellaneous Gems of Algebraic Combinatorics

Chapter
Part of the Undergraduate Texts in Mathematics book series (UTM)

## Abstract

An evil warden is in charge of 100 prisoners (all with different names). He puts a row of 100 boxes in a room. Inside each box is the name of a different prisoner. The prisoners enter the room one at a time. Each prisoner must open 50 of the boxes, one at a time. If any of the prisoners does not see his or her own name, then they are all killed. The prisoners may have a discussion before the first prisoner enters the room with the boxes, but after that there is no further communication. A prisoner may not leave a message of any kind for another prisoner. In particular, all the boxes are shut once a prisoner leaves the room. If all the prisoners choose 50 boxes at random, then each has a success probability of 1/2, so the probability that they are not killed is 2−100, not such good odds. Is there a strategy that will increase the chances of success? What is the best strategy?

## Keywords

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## References

1. 4.
L. Babai, P. Frankl, Linear Algebra Methods in Combinatorics, preliminary version 2 (1992), 216 pp.Google Scholar
2. 7.
E.R. Berlekamp, On subsets with intersections of even cardinality. Can. Math. Bull. 12, 363–366 (1969)
3. 11.
R.C. Bose, A note on Fisher’s inequality for balanced incomplete block designs. Ann. Math. Stat. 619–620 (1949)Google Scholar
4. 16.
Y. Caro, Simple proofs to three parity theorems. Ars Combin. 42, 175–180 (1996)
5. 21.
E. Curtin, M. Warshauer, The locker puzzle. Math. Intelligencer 28, 28–31 (2006)
6. 34.
R.A. Fisher, An examination of the different possible solutions of a problem in incomplete blocks. Ann. Eugen. 10, 52–75 (1940)
7. 43.
A. Gál, P.B. Miltersen, The cell probe complexity of succinct data structures, in Proceedings of the 30th International Colloquium on Automata, Languages and Programming (ICALP) (2003), pp. 332–344Google Scholar
8. 46.
R.L. Graham, H.O. Pollak, On the addressing problem for loop switching. Bell Syst. Tech. J. 50, 2495–2519 (1971)
9. 47.
R.L. Graham, H.O. Pollak, On embedding graphs in squashed cubes, in Lecture Notes in Mathematics, vol. 303 (Springer, New York, 1973), pp. 99–110Google Scholar
10. 68.
K.H. Leung, B. Schmidt, New restrictions on possible orders of circulant Hadamard matrices. Designs Codes Cryptogr. 64, 143–151 (2012)
11. 76.
J. Matoušek, Thirty-Three Miniatures (American Mathematical Society, Providence, 2010)
12. 95.
H.J. Ryser, Combinatorial Mathematics (Mathematical Association of America, Washington, 1963)
13. 102.
R. Stanley, Differentiably finite power series. Eur. J. Combin. 1, 175–188 (1980)
14. 108.
R. Stanley, Enumerative Combinatorics, vol. 2 (Cambridge University Press, New York, 1999)
15. 110.
K. Sutner, Linear cellular automata and the Garden-of-Eden. Math. Intelligencer 11, 49–53 (1989)
16. 114.
R.J. Turyn, Character sums and difference sets. Pac. J. Math. 15, 319–346 (1965)
17. 115.
R.J. Turyn, Sequences with small correlation, in Error Correcting Codes, ed. by H.B. Mann (Wiley, New York, 1969), pp. 195–228Google Scholar