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