# ISILC Proof Theory Symposion. Dedicated to Kurt Schutte on by J. Diller, G.H. Müller

By J. Diller, G.H. Müller

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Peters, on the faculty of the University of Rochester Medical School, discovered a general procedure for constructing maximum-length rooktours on square boards of any size. See his paper "Rooks Roaming Round Regular Rectangles," m Journal of Recreational Mathematics, Vol 6, 1973, pages 169-173. Frederick Hartmann of Rolling Hills Estates, Calif, extended the analysis to nonsquare rectangular boards, but so far as I know, his results remain unpublished. When the board h n X 1, it reduces to the "worst-route" problem for a postman delivering mail to a row of n houses (see my Sixth Book of Mathematical Games from Scientific American, W H.

For two symbols the formula giving the number of chains is where k is the number of symbols in a group. Any of the 2,048 chains provides a convenient way of recording the key to a biliteral cipher. Simply print the alphabet, with the first six digits appended to bring the number of symbols to 32, in a circle and add the chain of ^'s and b's inside the circle [see Figure 25]. To obtain the sequence for, say, R, check the set of five symbols that start at R and go clockwise (or the other way if you prefer) around the circle.

Since 1 to any poly power of 1 is 1, all these Sanctions and their derivatives, when graphed against x, cross one another at x = 1, and their values at 0 are the limits as x approaches 0. Grosse's notes, which already fill many volumes, lead into a lush jungle of unusual theorems as well as new classes of numbers. Up and down dipowers obviously are identical, but for all higher polypowers the two directions give different numbers. The triplet of 9's, for example, when calculated upward is a number of only 77 digits.