# A page of trivia

1. The game of Moo (pdf), otherwise known as Mastermind, is a simple game that could be used as an example for a computer course. It is sufficiently complex that a complete analysis can take many hours of cpu time to evaluate, even today, but small enough to be easily programmed when suitable value functions are thought up. This paper examines a few value functions. An odt version is also available.

2. The disjoint interval problem (pdf). Consider a sequence of points all in a given interval, such that the first two are in different halves of the interval; the first three are in different thirds, the first four in different quarters, etc. How long can the sequence be? There is an odt format document, too.

3. The Unique Pairs problem (pdf). Define a set, S, of n  non-negative integers such that all possible pairwise sums, i+j: (i,j) in S, i  ≠ j, are distinct. What is the smallest possible largest element of S? What is the smallest possible sum of all elements in S? These were found for small values of n  by an exhaustive computer-aided search. There is also an odt format document.

4. Giving change (pdf) is further work done on the question of whether the average number of coins needed to make monetary values up to one GB pound can be reduced by changing the denominations of coins used. There is an odt format version, too.

5. The Reve's Problem (pdf). In the traditional Towers of Hanoi, there are three pegs on one of which are heaped a number of discs of different diameters with no disc resting on a smaller one. These are to be transferred to another one moving only one disc at a time from one peg to another, and such that at no time during the moving process does a disc rest on a smaller one. H. E Dudeney, a well-known early 20th century English puzzlist, suggested the generalisation to any number of discs. There is an odt format document, too.

6. Pythagorean quadrilaterals (pdf) are convex quadrilaterals whose diagonals intersect at right angles and whose sides and diagonals all have integral lengths. This paper is an edited copy of a paper first published in the Journal of Recreational Mathematics, and describes a search for small such quadrilaterals (also known as kites) especially where all the four right triangles are primitive Pythagorean ones. There is an odt format document, too.

7. Notes on the “four digits problem” (pdf) in W.W. Rouse Ball's Mathematical Recreations and Essays. This is a note confirming the limits that he found and a table of the methods up to 312. There is also an odt format document. These results are based on my notes of 50 years ago.

8. There are also some questions and answers from the Journal of Recreational Mathematics.