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The Numbers Game is one of the games played on UK Television's Countdown. (We have no affiliation
with the television show, the production company or Channel 4.) crosswordtools.com hosts
a perfect solver for it.
- Q1. How is the game played?
- Q2. How do you use the solver?
- Q3. Is the solver perfect?
- Q4. How do the selection of numbers affect how easy or difficult the game is?
- Q5. What makes one solution "distinct" from another?
- Q6. How does the solver rate a solution for intuitivenes?
- Q7. How does the solver rate a problem for difficulty?
- Q8. Can you give some examples of difficult games?
- Q9. How many different games are there?
- Q10. Has any game ever come up twice?
- Q11. What future issues are there to look at?
The rules are pretty simple. Six smaller numbers are chosen at random (in range 1-10,25,50,75 and 100)
and a target number
is generated in the range 100 to 999. The players then have 30 seconds to get as close to the
target number as possible using only the smaller numbers and basic arithmetic operations (addition, subtraction, multiplication
and division). There is no obligation to use all the smaller numbers but no number may be used more than
once. The only other important rule is that all the steps to the solution
must be whole numbers. e.g. You cannot start by dividing 7 by 5 and then use 1.4 in subsequent steps. Only
(positive) integers are allowed. Points are awarded according to how close you can get.
The only control that the contestants have over the choice of game is to choose the number of larger numbers
(25, 50, 75 and 100) that form the six numbers to be used. (The top row contains the bigger numbers and
the contestant can specify which rows the six numbers are drawn from.)
Using the solver should be straight-forward. If you have a specific game you wish to solve you just
need to fill in the numbers and target and click on Find solutions. To generate a random game, click on one
of the other buttons to select the ratio of large to smaller numbers. You can then try to solve it
yourself and solve it when you want to see how it is done.
Yes. It is believed that the solver will always find a solution if one
exists and will always give the closest solution if no perfect solution exists. Although it isn't
possible to mathematically prove that a program works (with rare exceptions), we are very
confident that it does!
In addition the solver will produce all the distinct ways of getting the
answer if the problem is solvable and list them in order of intuitiveness.
These features and the way it can rates games by difficulty makes it the definitive source of
information on this game.
This is an interesting question that we have answered with a large statistical study of
random games and their solutions and by passing these games through the solver.
On the television show, the small numbers are selected at random from cards which are
face down on a table. According to Countdown - Spreading The Word the top row
contains the numbers 25, 50, 75 and 100 and the other rows contain the numbers 1-10 (two of each).
The distribution of the cards means that no small number can appear more than twice and no larger
number can be repeated at all.
The contestant selects six numbers from the 24 cards. As the contestant can choose which
rows they come from they can determine how many big numbers are used but have no control apart
from that over which numbers are selected. The target number is then selected at random by a computer
in the range 100-999.
In the study we performed, the types of games were: six numbers selected at random regardless of the
rows; one large number; two large numbers; three large numbers and four large numbers. The criteria
looked at were: percentage of games that were perfectly solvable; percentage of unsolvable games
that can be solved only one away from the target; average (median) number of different solutions
that exist for the solvable games and average (median) difficulty of the solvable games. Here are the
results:
| |
Percentage solvable |
Percentage of unsolvable which are 1 away |
Median number of distinct solutions per solvable game |
Median difficulty (solvable games) |
| Six small numbers |
84% |
66% |
9 |
70% |
| One large, five small |
98% |
90% |
19 |
45% |
| Two large, four small |
98% |
93% |
20 |
41% |
| Three large, three small |
94% |
90% |
17 |
48% |
| Four large, two small |
90% |
90% |
13 |
68% |
| Six random (rows ignored) |
94% |
71% |
17 |
50% |
The results show the importance of large numbers. Even a single large number has
a dramatic effect on the easiness of the game, the chances that it is solvable and the likely
number of different solutions.
The level of easiness peaks at two large numbers though the game with two large numbers
is only slightly easier than having one large number. One needs more than fifty games before one
is likely to find an unsolvable game with two large numbers while it just under one
in fifty with one large number. This fact surprises a lot of people.
Another difference is how far off the best solution can be when the game isn't solvable.
With one or more large numbers the vast majority of unsolvable games can be solved with
a target one away from the true target and games where the best solution is more than
a handful away are almost impossible to construct let alone find at random.
With six small numbers, sometimes the closest answer is
hundreds away from the correct answer. An extreme example would be (2,2,4,1,3,1) target: 959
whose best solution is an incredible 851 away from the target!
Another interesting thing that the solver reveals is how many different ways there are
of solving the games. This is something that doesn't come over when watching the television show.
The contestants do occasionally come up with different ways to the solution but the sheer volume
of solutions for most games will surprise most people.
From the above table you can see that even with six small numbers a typical solvable game
has nine different solutions. Some games have very
many more solutions than that. An extreme example would be something like
(75,8,2,10,6,4) target: 750
which has an incredible 355 distinct solutions in addition to the obvious one (75x10).
A more difficult example would be (75,4,1,9,7,3) target: 219 which has a mere hundred ways of
solving it!
Essentially the solver does the following:
It ignores certain categories of solution where operations are wasted.
These include:
- Solutions where something is multiplied by 1.
- Solutions where something is multiplied by 0.
- Solutions where something is divided by 1.
- Solutions where 0 is added to something.
- Solutions where an operation is done to create a new number but the new
number is not used in creating the final result.
All the above solutions are considered to be the same as the solution where these
pointless operations are not done.
It also considers two solutions to be the same if:
- They produce the same solution and
- They use exactly the same group of original numbers and
- The basic operation that is done to each original number is the same. These operations
include (1) addition (being on either side of an addition operation or the left of a subtraction);
(2) being multiplied; (3) being subtracted from something or being on the top (4) or bottom (5) of a division operation.
The solver also does not distinguish between identical original numbers. e.g. If there are
two 5's, a solution that uses the first five for one purpose and the second for another
is not distinguished from a solution that does it the other way around.
This definition does deal with most solutions that intuitively would be considered the same (e.g.
adding numbers together in a different order). However, there are some cases which arguably should
be considered distinct but are not. i.e. The solver might be merging some solutions into one another
which should be listed separately. An example would be (((50x2)x2)+(9x4)) and ((50x4)+(9x(2x2)))
which are currently considered the same.
Both cases use a 4 which is in the original numbers and one which is created by multiplying 2x2 so
it could also be argued that they are sufficiently similar to warrant being merged.
How the solver defines equality between solutions is currently the one weak point. We already have
an improved definition which appears to correspond more closely with human intuition. This new definition
will be implemented in due course.
Essentially the program embodies rules of thumb about the difficulty of certain
arithmetic operations. For example, adding 25 to 100 is certainly easier than multiplying 37 by 71 or
dividing 741 by 13. By combining (fairly subjectively chosen) weights to each arithmetic operation that
makes up a solution, the solver can create an overall intuition factor. This
score also embodies the number of operations done giving a bias towards solutions using fewer of the original
numbers. The solver also uses this function to choose the most intuitive solution for the various
solutions it considers equivalent.
The solver also uses this factor to sort distinct solutions into
order, most intuitive first. The solution at the bottom of the list should thus be the one that is
least likely to be found by a human!
The solver's scores for intuitiveness can never be perfect as the concept is subjective. However,
there is scope for some improvement. Again this is something that will be improved in due course.
It combines factors such as the intuitiveness of the most
intuitive solution and the number of different solutions to form a raw score. This raw
score is then turned into a percentile by comparing it with the results of thousands of
random games generated using the "six from anywhere" selection method. A percentile below
50% means that it is easier than an average game and above 50% means that it is harder.
As these percentiles are taken from the "six from anywhere" method they won't correspond
exactly with you big number choices. e.g. More than 50% of games where you select two large numbers
will have difficulty scores less than 50%
These games are drawn from the most difficult end of the scale. They are all solvable
but very, very difficult and illustrate how hard this game can be.
If you can solve these in 30 seconds or less each, you can consider yourself super-human
and should consider applying to replace Carol Vorderman (or at least appearing as a contestant on
the television show)!
Click the problem to see it in the solver.
- 25, 6, 3, 3, 7, 50 Target: 712
- 50, 2, 6, 4, 10, 4 Target: 687
- 8, 75, 8, 4, 6, 10 Target: 993
- 6, 2, 8, 7, 8, 4 Target: 917
- 7, 8, 50, 8, 1, 3 Target: 923
- 9, 6, 10, 4, 6, 2 Target: 946
According to the author's calculations there are 13243 possible
combinations of numbers that can be selected giving 11918700 possible
games (the order of the numbers is not considered important). This assumes that there are 900 targets in the range 100-999 inclusive. The issue of 100 being generated as a target number causes some problems.
If 100 is excluded as a target, there are 11905457 possible games.
The issue of whether CECIL can generate 100 as a target is a complicated one.
According to Countdown - Spreading The Word
it can be generated but emails from two regular viewers contradict each other. One says that he
has seen almost every one of the 2000+ shows and it has never come up. Another viewer claims that he has seen a show
where it appears. We also have a report a numbers game with 100 as a target being listed in a book of
published Countdown problems. Because of the book and because remembering something that didn't
happen seems less likely than forgetting something we believe that it can be generated but what happens when 100 is also picked as one of the
numbers to use to find the target? Nobody seems to know. Our guess is that if that did happen they would simply
not play the game, perhaps creating a new game to broadcast. This issue makes a precise count of the possible games
a little complicated
The calculation for the number of groups of non-target numbers is complicated by the duplicates.
Essentially, the combinations are partitioned into a number of groups depending on
what categories (small or large numbers) are in the group and how many duplicate numbers
there are. The number of combinations in each group is then calculated and these numbers summed up.
If lowercase letters represent the small numbers (1-10) and uppercase the large numbers
(25, 50, 75 and 100), the following are the possibilities.
According to Countdown - Spreading The Word
each of the smaller numbers is on two cards and each big number is only on one, limiting
the number of duplicates of a particular small number that can appear to 2.
3 duplicate numbers:
aabbcc : 10C3 = 120 possibilities
2 duplicate numbers:
aabbcA : 10C34C13C2 = 120x4x3 = 1440 possibilities
aabbcd : 10C44C2 = 210x6 = 1260 possibilities
aabbAB : 10C24C2 = 45x6 = 270 possibilities
1 duplicate number:
aabcde : 10C55C1 = 252x5 = 1260 possibilities
aabcdA : 10C44C14C1 = 210x4x4 = 3360 possibilities
aabcAB : 10C34C23C1 = 120x6x3 = 2160 possibilities
aabABC : 10C24C32C1 = 45x4x2 = 360 possibilities
aaABCD : 10C1 = 10 possibilities
All numbers different:
abcdef : 10C6 = 210 possibilities
abcdeA : 10C54C1 = 252x4 = 1008 possibilities
abcdAB : 10C44C2 = 210x6 = 1260 possibilities
abcABC : 10C34C3 = 120x4 = 480 possibilities
abABCD : 10C2 = 45 possibilities
120+1440+1260+270+1260+3360+2160+360+10+210+1008+1260+480+45=13243 possible combinations.
The above calculation also has the advantage of showing how many of each pattern there
are with the big numbers made explicit. For example, to work out how many combinations with one
big number there are you can sum just the possibilities for those cases.
A simpler way of getting just the total is to forget about big numbers and only look at
the number of duplicates. Having selected the numbers to be duplicated twice (from ten possibilities)
you can then select the remaining from the number of different numbers left. i.e.
3 pairs: 10C3 = 120
2 pairs: 10C212C2 = 2970
1 pair: 10C113C4 = 7150
0 pairs: 14C6 = 3003
These also sum to 13243 possibilities and is thus a nice double-check on the arithmetic above.
In all the above nCk is mathematical notation meaning "the number
of combinations of k items that can be selected from a list of n distinct items". The formula
for it is n!/(k!(n-k)!) where "!" is the factorial function (n!=nx(n-1)x(n-2)...x2x1).
As far as we are aware nobody has kept a definitive record of which numbers games have
come up in the show's very long history. The TV programme has been running since 1982 and
approximately 2500 programmes have been broadcast.
However, in the same way that two people having the same birthday in a small group
of people is surprisingly likely, the chances of two identical games having come up are remarkably high despite
there being more than a million possible games!
If we assume that 5000 numbers games have been broadcast, this implies that
there are 5000x4999/2 or 12497500 possible pairings of games, each of which have
a probability of 1/11905457 of being the same.
Therefore the chances of none of them being the same are (1-1/11905457)12497500
or 35%. (This assumes that the probabilities of each pair of games being the same is independent.
This isn't strictly true but is reasonable considering how large the number of possible
games is compared with the number played.) This means that there is a 65% chance that
at least one numbers game has been repeated exactly since the show started being broadcast!
In practice, the probability is higher because the above maths assumes that all games
are equally likely and in reality there is a bias towards games with the more popular counts of
large numbers (e.g. one large number) as this is within the control of the players.
1. Fractional steps. The rules of the game forbid using non-integer, rational numbers (i.e. non whole
numbers) as the intermediate steps towards a solution. Obviously this is only an issue with
the divide operation and even then it is usually possible to re-express the solution using only
integers. However, there are some games which have solutions with fractions along the way and no
integer solutions. It would be interesting to explore these and see how big a difference this rule
makes to the game.
2. Exponents. The rules also forbid putting one number to the power of another. e.g.
with 5 and 2 you cannot make 25 by squaring the 5 nor can you make
32 by putting 2 to the power of 5. It would be interesting to relax this
rule and see what effect it had on the solvability of the games.
Go to the Numbers Game Solver tool!
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