mscroggs.co.uk
mscroggs.co.uk

subscribe

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:
\(n\)
oddeven
\(m\)oddevenodd
eoddeven
Using only \(+\) and \(\times\), can you construct functions \(g(n,m)\) which give the following output tables:
\(n\)
oddeven
\(m\)oddoddodd
eoddodd
\(n\)
oddeven
\(m\)oddoddodd
eoddeven
\(n\)
oddeven
\(m\)oddoddodd
eevenodd
\(n\)
oddeven
\(m\)oddoddodd
eeveneven
\(n\)
oddeven
\(m\)oddoddeven
eoddodd
\(n\)
oddeven
\(m\)oddoddeven
eoddeven
\(n\)
oddeven
\(m\)oddoddeven
eevenodd
\(n\)
oddeven
\(m\)oddoddeven
eeveneven
\(n\)
oddeven
\(m\)oddevenodd
eoddodd
\(n\)
oddeven
\(m\)oddevenodd
eoddeven
\(n\)
oddeven
\(m\)oddevenodd
eevenodd
\(n\)
oddeven
\(m\)oddevenodd
eeveneven
\(n\)
oddeven
\(m\)oddeveneven
eoddodd
\(n\)
oddeven
\(m\)oddeveneven
eoddeven
\(n\)
oddeven
\(m\)oddeveneven
eevenodd
\(n\)
oddeven
\(m\)oddeveneven
eeveneven

Show answer & extension

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?

Show answer & extension

Tags: numbers, games
If you enjoyed these puzzles, check out Advent calendar 2024,
puzzles about advent, or a random puzzle.

Archive

Show me a random puzzle
 Most recent collections 

Advent calendar 2024

Advent calendar 2023

Advent calendar 2022

Advent calendar 2021


List of all puzzles

Tags

odd numbers chess functions numbers taxicab geometry indices unit fractions gerrymandering fractions dates pascal's triangle dominos crosswords time spheres probability irreducible numbers lines christmas complex numbers cube numbers ave sets clocks bases integration squares wordplay parabolas proportion rectangles multiples prime numbers addition arrows combinatorics dice planes scales surds integers factors digital clocks circles books grids games division graphs factorials crossnumbers triangle numbers star numbers consecutive numbers people maths palindromes square roots logic number hexagons colouring elections expansions geometry shapes albgebra digits calculus folding tube maps range sums perfect numbers quadratics geometric means products the only crossnumber shape cards powers partitions trigonometry percentages rugby 3d shapes medians sum to infinity axes regular shapes mean doubling sequences polygons multiplication cubics money decahedra algebra geometric mean floors volume means square numbers coordinates menace polynomials even numbers cryptic crossnumbers chalkdust crossnumber median square grids routes digital products neighbours pentagons dodecagons quadrilaterals ellipses cryptic clues sport triangles 2d shapes matrices tangents averages area numbers grids differentiation determinants advent remainders balancing perimeter chocolate angles binary tournaments tiling symmetry probabilty coins consecutive integers speed

Archive

Show me a random puzzle
▼ show ▼
© Matthew Scroggs 2012–2025