The easiest and hardest Countdown numbers games

I have been experimenting with exhaustive stats for the Countdown numbers game.

The question was simple: if we enumerate every legal set of six numbers, what can we say about the targets they can and cannot solve?

There are 13,243 distinct unordered number selections. This treats, for example, 1, 1, 2, 3, 4, 5 as one selection, not as several different physical ways to draw tiles. For probability questions, I weighted those selections by how many physical draws they represent from the Countdown numbers rack.

How difficulty is scored

The site's numbers solver has a rough difficulty score for ranking possible solutions to the same number selection and target.

The score favours short, round, mental-arithmetic-friendly calculations. Each arithmetic step is scored from the number produced by that step, with trailing zeroes stripped. Addition uses the base cost, subtraction is multiplied by 1.05, multiplication by 1.2, and division by 1.3. For multiplication by 10, the base cost is set to 1 before the multiplication weighting is applied, so the step costs 1.2 regardless of the size of the result.

The easiest game

The easiest possible game is a target that is already one of the numbers.

For example:

100, 75, 50, 25, 7, 3
Target: 100

The solution is just:

100 = 100

Using that difficulty score, this has difficulty 0, because it needs no arithmetic.

The easiest non-zero case is also very small:

25 + 75 = 100

The score for that step is 1. The solver scores a step from the number it makes, but removes trailing zeroes first, so making 100 has the same base cost as making 1.

That is why 25 + 75 = 100 is easier under this metric than something like:

100 + 1 = 101

The latter is still a trivial human calculation, but the step makes 101, so its difficulty score is 101. This odd edge case is not a practical problem: the score is only used to rank possible solutions for the same number selection and target, not to make official claims about human difficulty.

The hardest game I found

Using the solver's current difficulty metric, the hardest solvable standard target I found was:

Numbers: 7, 7, 25, 50, 75, 100
Target: 933
Difficulty: 74715.0

Try solving it before opening the answer.

7 + 7 = 14              difficulty 14
50 * 100 = 5000         difficulty 6
14 * 5000 = 70000       difficulty 8.4
70000 - 25 = 69975      difficulty 73473.8
69975 / 75 = 933        difficulty 1212.9

Which large-number choice is most solvable?

For each legal physical draw and each target from 100 to 999, I checked whether an exact solution exists.

Here are the probabilities, grouped by the number of large tiles chosen:

By this measure, 2 large gives the best chance of an exact solution.

That surprised me a little. The usual folklore is that one large is comfortable and four large is chaotic, which still seems fair. But if the question is just "does an exact solution exist?", two large numbers comes out fractionally ahead.

How hard are solvable games?

Among solvable targets from 100 to 999, the mean, standard deviation, and median difficulty look like this:

Again, this is weighted by physical draws. A distinct-looking set that can be drawn in more ways counts more often.

The median does not move very dramatically. The mean and standard deviation do, especially for four large numbers. That suggests most four-large games are not absurdly difficult, but there are some extreme outliers.

Targets outside the normal range

The televised game only uses targets from 100 to 999, but I also looked beyond that range.

Every target from 0 to 62,662 is solvable by at least one legal input selection.

The smallest target that cannot be solved by any legal selection is:

62663

The first ten unsolvable targets are:

62663, 65217, 71669, 73162, 77433, 77434, 77639, 77642, 77647, 77711

The smallest target that is solvable by exactly one distinct selection is:

58661

The only selection that solves it is:

3, 5, 6, 8, 75, 100

One solution is:

5 + 8 * (3 + 75) * (100 - 6) = 58661

The smallest non-universal target

At the other end, every target from 0 to 29 is solvable by every distinct legal selection.

The first target that is not universal is 30. It is solvable by 13,242 of the 13,243 distinct selections.

The one selection that cannot make 30 is:

1, 1, 2, 50, 75, 100

What to take from this

If you want the best chance of an exact solution, choose 2 large.

If you want reliable comfort, 1 large is almost as solvable and has the lowest standard deviation in this solver's difficulty score.

If you want volatility, choose 4 large. It still has exact solutions most of the time, but it produces the biggest difficulty outliers.

And if the target is already sitting in front of you, take the gift.