A page of trivia
 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.
 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.
 The
Unique Pairs problem (pdf).
Define a set, S, of n
nonnegative 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
computeraided search. There is also an
odt format document.

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.

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 wellknown early 20th century English puzzlist, suggested
the generalisation to any number of discs.
There is an
odt format document,
too.

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.
 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.
 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.
 There are also some
questions and answers
from the Journal of Recreational Mathematics.
 Here are some
results in Euclidean geometry with elementary proofs
which I have collected over the years.

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.
 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.
 Both Dudeney and the American Sam Loyd posed in their time puzzles based on finding a pair
of rationalsided 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.
 It is possible to express the n+1th 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)(a^{2}+b^{2})
– 2(a^{3}+b^{3}).
This
note (pdf) contains the expressions
for 2, 3 and 4 variables.
 I don‘t know who discovered this, but it was a question in
J.H. Conway‘s repertoire of oddities. Find a 10digit number,
all 10 digits different, such that 1 divides the first digit; 2 divides
the first 2 digits treated as a 2digit number; 3 divides the first
3 digits, 4 divides the first 4; etc, up to 10 divides the 10digit number.
You do not need a computer, as
this note (pdf) explains.
 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 fivepiece solution..
 In chapter 2 of his book “Tracking the Automatic Ant”, David Gale reports that
Lee Sallows had found this selfreferential sentence:
This computergenerated 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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