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M500 180 Page 25

think it to be irrational not to play this game, once, for a trillion pounds.

However, if you were to advise others to be as rational, or thought this

a good enough reason to play again, or others thought the same way inde-

pendently, then terrible things could happen. It is as if the introduction of

the mean concept into assumptions based on the mode concept of expec-

tation make the reasonable expectations for a whole set of games less than

the sum of the reasonable expectation of any one game. This can’t be true

(any more than the outcome of a fair toss can depend upon prior tosses) so

perhaps (to propose a ‘solution’ similar to the one I proposed in the case

of the surprise exam paradox) it is not, after all, rational to play this game

for a trillion (monopoly) pounds.

Problem 180.6 – Two pedestrians

Simon Rolph

Two pedestrians, P

and P

, are travelling at right angles to each other, but

are obscured by the corner of a building. Each is travelling parallel to the

wall nearest to them. Pedestrian P

is travelling at a constant speed of V

and the perpendicular distance between P

and the building is L

, i = 1, 2.

If the time they meet occurs at zero, ﬁnd the time before impact at

which they can both see each other across the corner of the building.

Solution 176.5 – Construct a square

Given a unit line segment, construct a unit square using com-

passes only.

Gordon Alabaster

Given unit line segment OA,

scribe a circle with centre O and

radius 1. With the compasses

set at 1, mark oﬀ from A to B,

from B to C and from C to D,

round the circle. By Pythago-

ras’ theorem, AC has length

√

Set the compasses to

√

3. Scribe

arcs from A and from D to cross

at E. By Pythagoras’ theorem,

OE =

√

Set the compasses to 1. Scribe arcs from E to intersect the circle with

centre O at F and G. Then the quadrilateral OFEG is a unit square since

OF = F E = EG = GO = 1 and OE =

√

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