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. A type of card shuffle known as “topswops” consists of reversing the order of the top N cards where N is the number showing on the top card. Martin Gardner introduced the question of what is the longest sequence of such shuffles before the top card becomes an ace, when it lapses into a loop of length 1. I wrote a paper (pdf) on the subject in 1989, which has a simple update here.

  8. 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.
    A similar problem is the use of four fours to make numbers. But he uses slightly different conditions for the use of operators. Here is further description of this work, including later solutions of the case of {1, 2, 3, 4} and other comments.

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

  10. Here are some results in Euclidean geometry with elementary proofs which I have collected over the years.

  11. In one of Martin Gardner's Mathematical Games columns for Scientific American, he asked about dividing a rectangle into 5 smaller ones, where every side was of integer length, and no two were equal. The layout is shown on the right.

    What is the smallest rectangle that can be divided under these conditions?
    Here (pdf) is one way to tackle it.

  12. Henry Earnest Dudeney was an English amateur mathematician who set puzzles for publication in magazines, and collected some into books. On a very few occasions he left a challenge for the reader unanswered. Here is one, to find three rational squares in arithmetic progression with a commn difference of 23. He provided answers, without any indication of how he found them, for the cases of a difference of 5, 7 and 13, but left 23 as an exercise. Here (pdf) is one way to tackle it.

  13. Both Dudeney and the American Sam Loyd posed in their time puzzles based on finding a pair of rational-sided cubes whose volumes add to a simple integer. Here is a comment on the problem (pdf), and a table of results for integers less than 100 which I found with a computer search.

  14. It is possible to express the n+1-th power of the sum of n variables as a combination of various powers of sums of powers. For instance (a+b)3 = 3(a+b)(a2+b2) – 2(a3+b3). This note (pdf) contains the expressions for 2, 3 and 4 variables.

  15. I don‘t know who discovered this, but it was a question in J.H. Conway‘s repertoire of oddities. Find a 10-digit number, all 10 digits different, such that 1 divides the first digit; 2 divides the first 2 digits treated as a 2-digit number; 3 divides the first 3 digits, 4 divides the first 4; etc, up to 10 divides the 10-digit number. You do not need a computer, as this note (pdf) explains.

  16. It is possible to dissect with straight line cuts an equilateral triangle into a few pieces, each of which is an isosceles triangle, no two of which are similar, and none are equilateral. What is the fewest number of pieces required? Here's a five-piece solution..

  17. In chapter 2 of his book “Tracking the Automatic Ant”, David Gale reports that Lee Sallows had found this self-referential sentence:

    This computer-generated sentence contains two hundred forty seven letters: four a's, one b, four c's, five d's, forty four e's, nine f's, three g's, seven h's, eleven i's, one j, one k, three l's, two m's, twenty nine n's, nineteen o's, two p's, one q, fourteen r's, thirty one s's, twenty five t's, seven u's, eight v's, seven w's, two x's, six y's, and one z.

    You will notice that it indeed correctly describes itself. Is this the only such sentence? Or are there any others? In fact, yes. (pdf)

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