Advent calendar 2020
6 December
There are 12 ways of placing 2 tokens on a 2×4 grid so that no two tokens are next to each other horizonally, vertically or diagonally:
|
|
|
|
|
|
||||||||||||||||||||||||||||||||||||||||||||||||
|
|
|
|
|
|
Today's number is the number of ways of placing 5 tokens on a 2×10 grid so that no two tokens are next to each other horizonally, vertically or diagonally.
Show answer
Hide answer
First, consider placing 5 tiles in a 1×9 grid. There is only one way to do this:
O | O | O | O | O |
To get the number of ways of placing 5 tiles in a 1×10 grid, imagine adding an extra blank square to either the start or end of the grid or between two of the counters.
There are 6 places this tile could be inserted leading to 6 arrangements of 5 tiles in a 1×10 grid.
For 5 tiles in a 2×10 grid, you can first pick the columns the tiles go in (as a tile being in a column means nothing can be placed the columns either side, the number of ways to pick
columns is the same and the number of wats to arrange 5 tokens in a 1×10 grid). For each of these column choices, there are two locations for each tile (top or bottom).
This leads to a total number of arrangements of 6×25=192.
Tags: numbers, combinatorics
Archive
Show me a random puzzle Most recent collections
Advent calendar 2023
Advent calendar 2022
Advent calendar 2021
Advent calendar 2020
List of all puzzles
Tags
chocolate taxicab geometry ellipses division menace geometry tangents dominos symmetry geometric means quadrilaterals consecutive numbers median sum to infinity fractions prime numbers crossnumbers expansions mean polygons range irreducible numbers angles unit fractions planes chalkdust crossnumber ave crossnumber square numbers routes logic tournaments graphs tiling means probabilty speed decahedra integration people maths gerrymandering volume number bases rugby probability pascal's triangle coordinates sums star numbers partitions 3d shapes remainders coins percentages time functions arrows determinants binary games cubics cards trigonometry addition digits multiplication cryptic crossnumbers dodecagons crosswords circles factorials shapes polynomials albgebra sport geometric mean quadratics books numbers perimeter digital clocks palindromes triangles regular shapes colouring algebra cryptic clues differentiation multiples digital products proportion money the only crossnumber balancing surds scales even numbers grids wordplay shape advent sets perfect numbers matrices area 2d shapes square roots pentagons odd numbers calculus averages factors folding tube maps dice complex numbers products rectangles lines parabolas squares chess spheres consecutive integers hexagons triangle numbers elections floors integers doubling axes combinatorics dates christmas sequences indices cube numbers clocksArchive
Show me a random puzzle▼ show ▼ ▲ hide ▲
Most recent collections
Advent calendar 2023
Advent calendar 2022
Advent calendar 2021
Advent calendar 2020
List of all puzzles
Tags
numbers time 3d shapes fractions gerrymandering coins triangle numbers median crossnumber tangents trigonometry division crosswords sequences folding tube maps unit fractions probability polygons multiples arrows square roots dominos crossnumbers grids factors clocks mean averages partitions chalkdust crossnumber cards logic multiplication range integration polynomials hexagons perfect numbers geometry albgebra cryptic clues tiling christmas ellipses advent complex numbers products combinatorics menace colouring taxicab geometry scales pascal's triangle prime numbers quadratics cube numbers proportion means parabolas books elections geometric means area the only crossnumber irreducible numbers matrices planes differentiation people maths square numbers geometric mean palindromes balancing factorials pentagons calculus star numbers sums tournaments perimeter chess indices triangles binary volume rugby angles number wordplay money sport chocolate floors determinants axes odd numbers cubics percentages cryptic crossnumbers coordinates circles spheres rectangles dates games surds routes integers remainders addition digits digital clocks expansions sets quadrilaterals digital products probabilty dodecagons sum to infinity regular shapes 2d shapes decahedra functions consecutive numbers symmetry lines speed shapes even numbers algebra ave graphs squares bases dice consecutive integers doubling shape© Matthew Scroggs 2012–2024