When exploring some triangle geometry, we stumbled upon a set of Cevian axes: lines having certain properties in common with the Euler line. These axes yielded interesting results that we had not seen before. However, we have since learnt that many of them were discovered in the 19th Century. We were also unaware that John Rigby had covered much of this ground in unpublished work in the 1990s (private communication). We are therefore reluctant to claim any of our results as new, even though we hope that some may be.

Interesting algebraic and geometric results can be obtained by interleaving integer sequences term by term. To introduce this we consider how a well-known sequence can be considered as being composed of subsequences.

The standard Fibonacci sequence (Fk):

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... (1)

is defined by the recurrence relation Fk = Fk - 1 + Fk - 2 for k ≥ 3 and F1 = F2 = 1.

The practical measurement of distance on land and sea has been of enduring interest for millennia. Here we outline a brief interaction between Francis Galton (1822-1911) and Arthur Cayley (1821-95) in which they too put the ‘geo’ back into geometry.

Galton wanted to measure everything. In the adventurous life of this many-sided Victorian scientist, quantification was the hallmark of his work. In measuring geographical distances Galton used an idea suggested by Sir George Everest (1790-1866), the Surveyor-General of India in the 1830s. In the pages of the journal of the Royal Geographical Society, Everest showed how to calculate the distance from one point to an inaccessible point. One application would be to measure the distance from a point on one side of a river to a point on the other side.

In an obituary of M. L. Urquhart in [1], David Elliott quotes him as claiming that Urquhart's theorem (below) is the most elementary theorem of Euclidean Geometry ‘since it involves only the concepts of straight line and distance’.

Urquhart's theorem

Let AC and AE be two straight lines.

Let B be a point on AC, D a point on AE, and suppose that BE and CD intersect at F.

If AB + BF = AD + DF then AC + CF = AE + EF. (1)

In the Name Game, letters of the alphabet are drawn at random, and a player marks off all occurrences of the letter in his name as it is called. The winner is the player whose name is deleted first; but a tie can occur when players' names have letters in common. For the two-player game, the probability of a player winning depends not only on the length of his own name but on how many letters occur only in the other player's name. (See [1] for probabilities involving more players.) For example, if Stephanie plays against Georges, then there are 11 letters in the union, 2 in the intersection, 3 that are in Georges but not Stephanie, and 6 that are in Stephanie but not Georges. In this case, the probability of a tie is 2/11, the probability of Stephanie winning is 3/11, and the probability of Georges winning is 6/11. These probabilities are easily derived by considering the 11! permutations of the letters in the union.

It has been known for some time that when the sun is reflected in a circular cylinder, the illuminated part of the cylinder's base is bounded by part of a nephroid (the curve obtained by rolling a circle around another circle with twice the radius).