Countdown Numbers Game

To find the solutions of a countdown number game, the only method is to make all possible calculations with the starting tiles.

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 gameshow with 6 number tiles (all natural integers not null), calculations use +, -, *, / operators but avoid non integer divisisions (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 type of algorithms for solving this number game:

Example: The numbers are 2, 5, 10

Recursive search: make all calculation 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 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