Sunday Afternoon Maths XXVI Answers & Extensions

This post contains the answers to this week's Sunday Afternoon Maths and some extension problems based around the originals.

Twenty-One

Virgil should go second. Whatever Scott adds, Virgil should then add to make four. For example, if Scott says 3, Virgil should say 1.

Using this strategy, Virgil will say 4, 8, 12, 16 then 20, forcing Scott to go above 21.

Extension

(i) If instead of 21, 22 cannot be said/beaten, how should Virgil win? How about 23? Or 24? How about \(n\)?

(ii) If instead of adding 1 to 3, 1 to 4 can be added, how should Virgil win? How about 1 to 5? Or 2 to 5? How about \(m\) to \(l\)?

(iii) Alan wants to join the game. Can Virgil win if there are three people? Can he win if there are \(k\) people?

Odd and Even Outputs

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

\(g(n,m)=1\)

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

\(g(n,m)=n\times m + n + m\)

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

\(g(n,m)=n\times m +n+1\)

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

\(g(n,m)=m\)

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

\(g(n,m)=n\times m+m+1\)

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

\(g(n,m)=n\)

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

\(g(n,m)=n+m+1\)

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

\(g(n,m)=n\times m\)

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

\(g(n,m)=n\times m+1\)

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

\(g(n,m)=n+m\)

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

\(g(n,m)=n+1\)

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

\(g(n,m)=n\times m+n\)

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

\(g(n,m)=m+1\)

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

\(g(n,m)=n\times m+n\)

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

\(g(n,m)=n\times m+n+m+1\)

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

\(g(n,m)=2\)

Extension

Can you find functions \(h:\mathbb{N}\times\mathbb{N}\times\mathbb{N}\rightarrow\mathbb{N}\) (call the inputs \(n\), \(m\) and \(l\)) to give the following outputs:

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

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

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

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

etc