Tool to solve countdown numbers games automatically, e.g. to find a given number by performing operations on other numbers.

Countdown Numbers Game - dCode

Tag(s) : Number Games

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The Numbers Round in the TV Game Show Countdown is a mathematical game whose objective is to get a number with the four elementary operators (+, -, ×, ÷) and a list of randomly picked numbers.

To find the solutions of a countdown number game, the only method is to make all possible math calculations with the starting tiles (dCode uses this method).

The general principle is to start with the list of `N` numbers, pick `2` and make all operations with these two numbers, if the result is the expected total, note the calculation as possible solution, else, store the result in the list and try again with the `N-1` new numbers in the list, and so on.

__Example:__ Numbers `2,6,10`, then for each couple among `(2,6),(2,10),(6,10)`, let's take `(2,6)`, make the calculations `2+6=8`, `2*6=12`, `6-2=4` and `6/2=3` to get 4 new numbers `(8,12,4,3)` that will make 4 new couples with the remaining `10`: `(8,10)`, `(12,10)`, `(4,10)` and `(3,10)`. Start over with new couples recursively.

The **original solver** uses the rules of the TV Countdown game show with 6 number tiles (all natural integers not null), calculations use `+`, `-`, `*`, `/` operators but avoid non integer divisions (leading to decimal numbers).

__Example:__ Find `123` with `4,5,6,7,8,9`. In 3 operations:`8 + 5 = 13; 13 x 9 = 117; 117 + 6 = 123`.

The **advanced solver** allows more options, constraints on operators, number of operations, etc. It also proposes to generate a list of all possible results from given numbers.

__Example:__ Find `24` by using all numbers `5,5,5,1` and allowing non integers values:`1 / 5 = 0.2; 5 - 0.2 = 4.8; 5 x 4.8 = 24`

The **N-Numbers solver** uses original rules but with any quantity of numbers. Given result is not the easiest one, a random one. Calculation can be very long, billions of iterations, and if there is no answer, it will never end.

There are three main types of algorithms for solving this number game:

__Example:__ The numbers are 2, 5, 10

Recursive search: make all calculations with N numbers. It uses 2 numbers and for each operation, retry with the results and the N-2 remaining numbers.

__Example:__ Take 2 and 5, make an addition: 2+5=7, a subtraction, 5-2 = 3, etc. Take the result 7 (ou 3), and unused numbers: 10, and start over. 7+10 = 17, etc.

Search with cache: same as the previous one, but stores the calculation to avoid remake them, so, slightly faster, but necessitate a lot of memory.

__Example:__ Take 2 and 5, see if results are already known for addition 7, and subtraction 3, etc. Go on more quickly.

Random search: can find a solution quickly but do not make all calculations, it can prove that a solution exists, but not that a solution does not exist.

__Example:__ Take 2 random numbers 5 and 10, make a random operation: multiplication, store the result: 5*10=50 and go on with remaining results: 2 and 50

Negative numbers are mostly ignored because they do not influence resolution. Indeed apply the operator `-` (minus) in front of any negative number to make it positive.

__Example:__ Numbers `3,2,-1` and target `7`, calculate `3*2-(-1)=7`

__Example:__ Numbers `3,2,1` and target `7`, calculate `3*2+=7`

The physical versions of the game have 24 tiles

1 to 10 | 2 copies |

25, 50, 75 and 100 | 1 copy |

Several mathematical problems are inspired by the account is good:

The sequence 1, 2, 4, 10, 29, 76, 284, 1413, 7187, 38103, 231051, 1765186, 10539427 here is defined by a(1)=1 and a(n) the smallest positive integer that cannot be obtained from the integers 0 to n-1, using each number at most once and the operators +, -, ×, and /.

__Example:__ For n=4, the authorized numbers are 0, 1, 2 and 3 and it is possible to find 4=1+3, 5=2+3, 6=2*3, 7=2*3+1, 8 =(1+3)*2, 9=(1+2)*3, but impossible to find 10, so a(4) = 10.

The sequence 1, 2, 4, 11, 34, 152, 1007, 7335, 85761, 812767 here is defined by a(1)=1 and a(n) the smallest positive integer which cannot be obtained from the integers a(i) with i < n-1.

__Example:__ For n=4, the authoritative numbers are 1,2,4, it is possible to find 5=1+4, 6=2+4, 7=1+2+4, 8=2*4, 9=2 *4+1, 10=(1+4)*2, but impossible to find 11, so a(4) = 11.

The sequence 1, 2, 4, 11, 34, 152, 1143, 8285, 98863, 1211572 here is similar but allows non-integer intermediate results.

Drawing 1, 2, 4, 11, 34, 152 allows players to find all solutions from 1 to 1006.

dCode retains ownership of the "Countdown Numbers Game" source code. Except explicit open source licence (indicated Creative Commons / free), the "Countdown Numbers Game" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Countdown Numbers Game" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Countdown Numbers Game" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!

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Cite as source (bibliography):

*Countdown Numbers Game* on dCode.fr [online website], retrieved on 2024-06-16,

- 6 Numbers Game Solver (Original)
- 6 Numbers Game Solver (Improved)
- N-Numbers Game Solver
- What is the countdown numbers game? (Definition)
- How to solve a countdown numbers game?
- Which solver to use?
- What are the most common algorithms for countdown games?
- How to use negative numbers?
- What is the distribution of tiles / numbers?
- How to play Countdown Number Game at school?
- What is the Countdown problem?
- Which set of 6 numbers has a solution for all calculations from 1 to 999?

countdown,count,calculation,addition,subtraction,multiplication,division,operation,1,2,3,4,5,6,7,8,9,10,25,50,75,100,mathematical,math,find,number

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