Countdown Numbers Game

The Numbers Round in the TV Game Show Countdown is a mathematical game whose objective is to get a target number (generally between 101 and 999) with the four elementary operators (+, -, ×, ÷) and a list of randomly picked numbers.

To find the solutions to a random number, the only method is an exhaustive search that tests all possible combinations. The number of calculations increases exponentially, making a complete exploration expensive for a computer and difficult to perform mentally.

To speed up manual calculations, two heuristics are particularly useful:

— approaching the target result using multiplication, then adjusting with smaller numbers by addition or subtraction;

— decomposing the target number, or a close number, into two or three factors to guide the operations.

These methods do not guarantee finding a solution, but they significantly reduce the number of possibilities to examine.

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. The computation time can explode (billions of iterations) and, if no solution exists, the search never ends.

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. This method explores the entire tree of possible calculations but presents an exponential complexity.

Search with cache: same as the previous one, but stores the calculation to avoid remake them. This speeds up the search but consumes more memory.

Random search (or Monte Carlo): 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.