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
games consecutive integers combinatorics floors geometric mean rugby christmas multiplication algebra geometric means tournaments cryptic crossnumbers pascal's triangle the only crossnumber palindromes digital products folding tube maps money taxicab geometry sport quadrilaterals cards graphs sets indices square numbers chocolate routes balancing geometry spheres people maths books pentagons addition scales cubics factorials clocks bases chess determinants prime numbers fractions squares unit fractions menace crossnumber consecutive numbers dates colouring number logic multiples volume functions wordplay sequences remainders triangles axes complex numbers doubling albgebra trigonometry integration chalkdust crossnumber quadratics perimeter polygons coins 3d shapes factors ave ellipses digital clocks square roots lines elections binary decahedra arrows means range cryptic clues sums area shape 2d shapes hexagons crosswords coordinates circles partitions parabolas angles grids dice averages dodecagons speed cube numbers dominos odd numbers regular shapes symmetry planes calculus advent polynomials time irreducible numbers matrices rectangles sum to infinity perfect numbers median numbers even numbers probabilty expansions tangents triangle numbers integers surds division digits probability mean percentages products crossnumbers star numbers gerrymandering proportion shapes differentiation tilingArchive
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
volume chess integers palindromes dice numbers christmas games division complex numbers partitions colouring cryptic crossnumbers odd numbers balancing perfect numbers area bases scales differentiation coordinates probability axes graphs sport regular shapes binary irreducible numbers ellipses combinatorics taxicab geometry polynomials even numbers shapes advent sums dominos money spheres coins multiples decahedra tangents number chocolate shape unit fractions angles expansions lines tiling polygons algebra crosswords crossnumber square roots hexagons grids factors time arrows albgebra geometry 3d shapes averages consecutive numbers floors elections 2d shapes digital products rugby fractions wordplay digits dodecagons pascal's triangle perimeter routes mean speed functions circles indices crossnumbers sets tournaments products gerrymandering multiplication trigonometry probabilty the only crossnumber folding tube maps matrices addition percentages factorials prime numbers surds quadrilaterals chalkdust crossnumber square numbers sum to infinity integration people maths clocks means squares triangle numbers parabolas doubling star numbers remainders triangles symmetry calculus cubics dates planes pentagons cube numbers consecutive integers range quadratics determinants rectangles geometric means geometric mean sequences median ave logic cryptic clues books cards menace proportion digital clocks© Matthew Scroggs 2012–2024