Sunday Afternoon Maths XXVI

Posted on 2014-08-24

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
\(m\)	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
\(m\)	e	odd	odd

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Tags: functions

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?

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Tags: numbers, games

If you enjoyed these puzzles, check out Advent calendar 2025,
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Sunday Afternoon Maths XXVI

Odd and even outputs

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Twenty-one

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