The elementary theorem that, if the triangles ΩAB, ΩPQ are directly similar, so also are the triangles ΩAP, ΩBQ, can be elevated into a general principle:

Given a point Ω and any number of points P1, P2,.. lying on a curve Γ, let Q1, Q2, … be points such that the triangles ΩP1Q1, ΩP2Q2, .. are all directly similar. Then Q1, Q2, … lie on a curve similar to Γ, obtainable by rotating Γ through the angle PrΩQr and applying the scale-factor ΩQr/ΩPr.

The practice of long jumping a motor cycle from an inclined ramp over obstacles such as parked buses has become popular with stunt-men. There have been serious accidents, some with fatal results. Some crashes have been caused by the faulty technique of keeping the throttle wide open after leaving the take-off ramp. In some cases this can lead to a backward rotation of the cycle by up to 90° during the jump, with inevitable disaster on landing.

In this paper we analyse a game that was published in the 1996 Burger King Kid’s Club calendar. This calendar included an activity for each month, and for the month of September the activity was called the Mountain Bike Rally. To begin the game, each player places a marker on day 1. A turn consists of each player flipping a fair coin. Each player whose coin comes up heads moves his marker forward one day. If a player’s coin shows tails, then he moves his marker back one day, unless his marker is on day 1, in which case it stays there. The object is to be the first player to reach square 31. We never played the game because it was clear that it would take a long time to complete.

There are numerous arithmetical puzzles concerning the quantities of beer that can be measured if the only available measures are a 3-pint jug and a 5-pint jug, or the exact amounts of money obtainable using only 3p and 5p coins, or the amounts that can be made with only 3p and 5p stamps.

David Hilbert, one of the giants of mathematics, delivered a lecture at the International Congress of Mathematics at Paris in 1900. The first part of the lecture, a preamble to his announcement of the now-famous 23 problems, began with the words:

Every geometrical truth has an individual beauty which delights the mind: but the most fascinating theorems, as an eminent mathematician rightly asserts, are those which have simplicity, depth, and obvious generality I propose to show that nearly all the important circles of the geometry of the triangle—the in-circle, the nine-points circle, the circum-circle, Taylor’s circle, Lemoine’s circle, and many other useful but nameless circles—are each and all particular cases of one fundamental general circle.

There is grave difficulty in showing elementary school students how mathematics is applied in practical situations (at least in Canadian schools). Since current emphasis is on pure mathematics, applied mathematics (and applying mathematics)—procedures of model building, approximation, analytical manipulation and interpretation—may remain obscure or unknown to students leaving school despite essential simplicity of principles.

Thus it is important to keep a look-out for problems which have an obvious practical interest or economic moment, but which can be tackled with the most elementary mathematics. Emphasis on the strategies of model building in the schools seems all the more sensible (and feasible) now that automatic numeration is permitting easy command of even complicated analyses.