Sunday Afternoon Maths XXVI

Here's this week's collection. Answers & extensions tomorrow. Why not discuss the problems on Twitter using #SundayAfternoonMaths or on Reddit.

It's no late to get tickets for EMF Camp, where I will be giving a talk on flexagons, folding tube maps and braiding.

Twenty-One

Scott and Virgil are playing a game. In the game the first player says 1, 2 or 3, then the next player can add 1, 2 or 3 to the number and so on. The player who is forced to say 21 or above loses. The first game went like so:

Scott: 3
Virgil: 4
Scott: 5
Virgil: 6
Scott: 9
Virgil: 12
Scott: 15
Virgil 17
Scott: 20
Virgil: 21
Virgil loses.

To give him a better chance of winning, Scott lets Virgil choose whether to go first or second in the next game. What should Virgil do?

Odd and Even Outputs

Let \(g:\mathbb{N}\times\mathbb{N}\rightarrow\mathbb{N}\) be a function.

This means that \(g\) takes two natural number inputs and gives one natural number output. For example if \(g\) is defined by \(g(n,m)=n+m\) then \(g(3,4)=7\) and \(g(10,2)=12\).

The function \(g(n,m)=n+m\) will give an even output if \(n\) and \(m\) are both odd or both even and an odd output if one is odd and the other is even. This could be summarised in the following table:

		\(n\)
		odd	even
\(m\)	odd	even	odd
	e	odd	even

Using only \(+\) and \(\times\), can you construct functions \(g(n,m)\) which give the following output tables:

\(n\) odd even \(m\) odd odd odd e odd odd	\(n\) odd even \(m\) odd odd odd e odd even	\(n\) odd even \(m\) odd odd odd e even odd	\(n\) odd even \(m\) odd odd odd e even even
\(n\) odd even \(m\) odd odd even e odd odd	\(n\) odd even \(m\) odd odd even e odd even	\(n\) odd even \(m\) odd odd even e even odd	\(n\) odd even \(m\) odd odd even e even even
\(n\) odd even \(m\) odd even odd e odd odd	\(n\) odd even \(m\) odd even odd e odd even	\(n\) odd even \(m\) odd even odd e even odd	\(n\) odd even \(m\) odd even odd e even even
\(n\) odd even \(m\) odd even even e odd odd	\(n\) odd even \(m\) odd even even e odd even	\(n\) odd even \(m\) odd even even e even odd	\(n\) odd even \(m\) odd even even e even even