ONE TO NINE by Andrew Hodges

Notes, links, updates and answers for Chapter Two.

Page 44: There is a great deal in the Blood of the Isles about what can be learnt from Y-chromosomes and mitochondrial DNA about the population of the British Isles.

Answers to problems

Page 40: Define (A NOR B) to mean 'neither A nor B is true'. Then A NOR A means the same as NOT A.
Then (A NOR A) NOR (B NOR B) means 'neither A nor B is false', which is the same as (A AND B).
(A NOR B) NOR (A NOR B) means 'It is not true that neither A nor B is true', which is the same as (A OR B).

Page 58: (0, 2) has the property (0, 2) × (0, 2) = (0 × 0 − 2 × 2, 0 × 2 + 0 × 2) = (−4, 0). So it is a square root of (−4). So is (0, −2).

You are unlikely to find a square root of (0, 1) by guessing. It needs a little algebra. Suppose it is (a, b).
Then we must have (a, b) × (a, b) = (0, 1), i.e. (a² − b², a × b + a × b) = (0, 1)
i.e. a² = b² and a × b = 1/2
So a = b and a² = 1/2,
so a = b = 1/√2, or a = b = − 1/√2.

Answers to problems

Page 52: The key to this puzzle lies in the asymmetry in the descriptions of the vertical and horizontal directions. 'Up' has a physical meaning, i.e. pointing towards the sky. But 'left' and 'right' are defined more abstractly. Try defining your 'right' hand by a physical description, e.g. 'the hand with the ring on' or 'the hand pointing towards the door'. Then the mirror leaves this property unchanged, just as it leaves 'up' unchanged.

Page 67: I shall try to add a photograph of this experiment...

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