Glove Problem

Let there be $m$ doctors and $n\leq m$ patients, and let all $mn$ possible combinations of examinations of patients by doctors take place. Then what is the minimum number of surgical gloves needed $G(m,n)$ so that no doctor must wear a glove contaminated by a patient and no patient is exposed to a glove worn by another doctor? In this problem, the gloves can be turned inside out and even placed on top of one another if necessary, but no ``decontamination'' of gloves is permitted. The optimal solution is

$$g(m,n)=\cases{ 2 & $m=n=2$\cr {\textstyle{1\over 2}}(m+1) ... ...r 2}}(m)+{\textstyle{2\over 3}}n}\right\rceil & otherwise,\cr}$$

where $\left\lceil{x}\right\rceil$ is the Ceiling Function (Vardi 1991). The case $m=n=2$ is straightforward since two gloves have a total of four surfaces, which is the number needed for $mn=4$ examinations.

References

Gardner, M. Aha! Insight. New York: Scientific American, 1978.

Gardner, M. Science Fiction Puzzle Tales. New York: Crown, pp. 5, 67, and 104-150, 1981.

Hajnal, A. and Lovász, L. ``An Algorithm to Prevent the Propagation of Certain Diseases at Minimum Cost.'' §10.1 in Interfaces Between Computer Science and Operations Research (Ed. J. K. Lenstra, A. H. G. Rinnooy Kan, and P. van Emde Boas). Amsterdam: Matematisch Centrum, 1978.

Orlitzky, A. and Shepp, L. ``On Curbing Virus Propagation.'' Exercise 10.2 in Technical Memo. Bell Labs, 1989.

Vardi, I. ``The Condom Problem.'' Ch. 10 in Computational Recreations in Mathematica. Redwood City, CA: Addison-Wesley, p. 203-222, 1991.