Number Countdown

21st May 2012

Our latest addition to Pencil and Paper Games is Number Countdown, a challenging game for those who like mental arithmetic.

How to play

Here's how our pencil-and-paper version of Number Countdown works.

One player secretly writes two large numbers (25, 50, 75, or 100) and four small numbers (from 1 to 10) on a piece of paper. The other player secretly writes a three-digit target.

After revealing the numbers and target, both players then have to try and get as near to the target as possible by combining some or all of the six numbers using addition, subtraction, multiplication, and division. Only integers may be used at any stage in the calculation, and you can only use each number once.

The version on Pencil and Paper Games allows you to provide the numbers or the target, or neither, and the computer will fill in the gaps with a random selection. You can also provide both numbers and target to get the computer to solve a Number Countdown problem.

Example

Here's a sample selection of numbers:

and a target:

Here's one possible solution:

Is the computer perfect?

Given enough time the computer can find the best solution to any Number Countdown problem. However, on Pencil and Paper Games the computer is restricted to returning the best solution found in one second, so it's possible for a human player to beat it.

Try the following two fiendishly difficult Number Countdown problems, each of which has an exact solution that won't be found by the computer player.

If you can solve these consider yourself a master player! The solutions will be given in a future blog post.

Problem 1 (+ - * and /)

Problem 2 (+ - * and /)

Advanced game

If you're feeling like an extra challenge Pencil and Paper Games allows you to include the "^" operator (raise to the power) by selecting the Allow ^ ? checkbox.

Some positions can only be solved exactly with the "^" operator. For example:

This cannot be solved exactly with the four standard operators, but can be solved quite easily using "^" as follows:

Problem 3 (+ - * / and ^)

Here's a third fiendish problem that can only be solved exactly with the help of the "^" operator:

You should be able to get to 1000 quite easily without ^, but even with ^ getting to 999 is extremely tough. 

References

If you are interested in seeing how the computer program works see Lisp in Small Parts - Number Countdown.

For a description of the British game show Countdown see the Wikipedia article Countdown (game show).

For an excellent analysis of Number Countdown by William Tunstall-Pedoe see Numbers Game Solver FAQ.