In a recent article in the Gazette (March 1983) Chris Haines described the problems which arose when he was arranging a tenth birthday party for his son. There were seven boys in addition to his son at the party, making eight all together. In order to prevent the party from getting too boisterous, Mr Haines decided to split the party into pairs who would play various boxed games for two players: for example, halma, draughts, snakes and ladders. The plan was to have a sort of tournament in which:

1. (1) each child would play every other child once and once only;

2. (2) in a given session, different games would be played by each pair as only one set of each game was available;

3. (3) each child would play as many of the available games as possible.

The solid which is the three-dimensional analogue of the plane quadrilateral appears to have been neglected by geometers, possibly on account of the difficulty of visualising it adequately from a plane figure. A model, however, makes an interesting object and is readily constructed from thin stiff wire and thread. The extraordinary number of collinearities, which are its principal feature, then become at once obvious. For the benefit of any one who wishes to construct such a model I would suggest the desirability of commencing with the external diagonal triangle PQR and the line P′Q′R′ (see Figs. 1 and 2), as otherwise it is difficult to get a compact model.

This paper presents some new theorems about the Pascal points of a quadrilateral. We shall begin by explaining what these are.

Let ABCD be a convex quadrilateral, with AC and BD intersecting at E and DA and CB intersecting at F. Let ω be a circle through E and F which meets CB internally at M and DA internally at N. Let CA meet ω again at L and let DB meet ω again at K. By using Pascal’s theorem for the crossed hexagons EKNFML and EKMFNL and which are circumscribed by ω, the following results can be proved [1]:–

1. (a) NK, ML and AB are concurrent (at a point P internal to AB)

2. (b) NL, KM and CD are concurrent (at a point Q internal to CD)