Countdown Numbers Game

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.

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).

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.

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:

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

The physical versions of the game have 24 tiles

The game is ideal for learning how to do calculations in school classes. The board game is available here (affiliate link)

A dedicated variant is the Krypto board game.

dCode provides answers for all calculations and game variants.

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 /.

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.

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.