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.
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.
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.
Negative numbers are 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|
Reminder : dCode is free to use.
The copy-paste of the page "Countdown Numbers Game" or any of its results, is allowed as long as you cite dCode!
Cite as source (bibliography):
Countdown Numbers Game on dCode.fr [online website], retrieved on 2022-11-28,