When I was growing up in the UK we had just three TV stations. These were BBC1, BBC2 and ITV.
Then, when I was 15, a new station was launched with imaginative name of Channel 4. The first program to air on this new channel was a game show called Countdown. Countdown is still shown today and is one of the longest running game shows in the World. At the time of writing this article, over 5,000 episodes have been broadcast. The show was not entirely original in concept, being based on a slightly earlier game shown in France called Des chiffres et des lettres (Numbers and Letters). The Countdown game tests contestants' numeracy and wordmaking skills. In the ‘letters’ rounds of the game, players have to create the longest word possible from a random selection of letters (a little bit like Scrabble) picked from vowels and consonants. 
In the ‘numbers’ round of the game, players have to combine six selected numbers (using just the four basic arithmetic operators) to get as close as possible to a randomly generated total. It’s is the numbers game that I’m going to talk about in this post. 
All games require rules, here are the rules for the numbers game of Countdown:
Six, facedown, numbered tiles are selected from twentyfour shuffled tiles.
The tiles are arranged into two groups: Large Numbers and Small Numbers.
There are four numbers in the large set { 25 , 50 , 75 , 100 }
There are twenty numbers in the small set, two each of the numbers 110
{ 1 , 1 , 2 , 2 , 3 , 3 , 4 , 4 , 5 , 5 , 6 , 6 , 7 , 7 , 8 , 8 , 9 , 9 , 10 , 10 }
One contestant selects as many numbers as desired (unseen) from the large set (between none and all four), and the balance are pulled from the small set to make six numbers in total.
A random threedigit target number is then chosen by a computer*.
The contestants are given 30 seconds to get as close as possible to the chosen target by using just the four basic arithmetic operators +  × ÷
Not all the digits need to be used.
Concatenation of the digits is not allowed (You can’t use a “2” and “2” to make “22”).
At no intermediate step in the process can the current running total become negative or involve a fraction.
Each numbered tile can only be used once in the calculation.
10 points are awarded for correctly getting the exact solution.
7 points are awarded for getting within 5 of the required solution.
5 points are awarded for getting within 10 points of the required solution.
*There is some speculation as to whether 100 is a possible target number generated by the computer. Some say that only numbers range 101999 are generated (which are the rules in the French variant), and other say that any three digit numbers is possible. I’m going to assume the latter in my calculations and say that a target of 100 is possible.
From the way the numbers are selected it’s clear that there can never be more than two of the same number. Also, there can only be one each of the larger numbers. If all four of the large numbers are taken then it’s assured that { 25 , 50 , 75 , 100 } will occur.
Here's an example of the game in action. In this instance, one number was selected from the large set, and the rest from the small set. { 50 , 8 , 3 , 7 , 2 , 10 } The randomly selected target was 556. 
There are multiple ways to solve this. The smallest solution requires just four numbers:
(50 × 10) + (8 × 7) = 556
A example of a more complicated solution is this:
(((50  7) × 3) + 10) × 8) ÷ 2 = 556
On the show, contestants' declared solutions are confirmed by one of the hosts (who then goes on to reveal a better/alternate solution if one of the contestants does not get an exact solution). For 26 years, this host role was performed by the talented Carol Vorderman. Being a numbers geek, I wanted to know many things:

Carol Vorderman  Countdown Numbers Host Image: ITV 
This was a fun piece of code to write. I elected to use a bruteforce approach. I worked out all possible combinations of ways the numbers/symbols could be arranged and, if they gave a valid solution, what that solution was.
First of all I needed to generate all unique combinations/subsets of numbers from the starting tiles. There are 13,243 distinct ways of picking the numbers following the rules. Then, for each of these distinct subsets, the number of distinct ways these numbers can be arranged in order was determined.
For each of these, all combinations of the operators (five in total) of +  × ÷ needed to be inserted between each number. The most obvious way to process these was to use Reverse Polish Notation to store the calculation (postfix) as opposed to the more traditional (infix).

A trap that needs to be avoided is the ordering the operators should be applied. For example 10(72) gives a very different solution to (107)2, just as (10×4)2 is very different from 10×(42). In RPN notation these distinctions are obvious (See table). 
Calculating all possible solutions is easy using RPN by permuting the order operators and operands (within the constraints of a stack, so some combination as not possible; you can't operate on numbers if there are none in the stack!)
An added benefit of using RPN is that, at the end of any operation you can check the stack and that will give you a possible intermediate solution (remember not all numbers need to be used for a valid solution).
As an example of this, in the example on the last line of the table, the first intermediate result is (1005)=95. This is not a valid answer, as it is less than a hundred (the lowest possible target). However the next operation is multiplication by 5. This gives a result of 425, which is valid. So, we already have a short (efficient) solution for 425 which is 5×(1005). We also have an intermediate solution of 416 for the next step (5×(1005))9 …
There are some obvious optimisations that can be made in the algorithm (plus a couple of things to watch out for):

I find it easier to interpret infix rather than postfix equations, so I wrote a function to convert these back and insert the appropriate parenthesis after the calculations were over.

100 = 2×50  101 = (2×(7×9))25  102 = ((2+9)×7)+25  103 = ((2+9)×10)7 
104 = (2×7)+(9×10)  105 = 7×(2510)  106 = 2×((7×9)10)  107 = (2×50)+7 
108 = (2+10)×9  109 = (2×50)+9  110 = (2+9)×10  111 = (2×(9+50))7 
112 = 7×(259)  113 = (7×9)+50  114 = 2×(7+50)  115 = (9×10)+25 
116 = (2×(7×9))10  117 = ((2+9)×10)+7  118 = 2×(9+50)  119 = 7×((92)+10) 
120 = 2×(10+50)  121 = (2×((7×9)+10))25  122 = 2×((7×10)9)  123 = (2×(7+50))+9 
124 = (2×(7+50))+10  125 = (72)×25  126 = 2×(7×9)  127 = ((2+9)×7)+50 
128 = (7+9)×(102)  129 = (2×(10+50))+9  130 = 2×((9×10)25)  131 = ((2+7)×9)+50 
132 = 2×(7+(9+50))  133 = 7×(9+10)  134 = ((72)×25)+9  135 = 9×(2510) 
136 = (2×(7×9))+10  137 = 2+(9×(2510))  138 = 2×(9+(10+50))  139 = (2×(7+50))+25 
140 = 2×(7×10)  141 = (2×(25+50))9  142 = 2+((9×10)+50)  143 = (2×(9+50))+25 
144 = (2+7)×(259)  145 = (2×(10+50))+25  146 = 2×((7×9)+10)  147 = (2+(9+10))×7 
148 = ((107)×50)2  149 = (2×(7×10))+9  150 = 2×(25+50)  151 = (2×(7×9))+25 
152 = (7×(252))9  153 = (7+10)×9  154 = (2×7)+((9×10)+50)  155 = 2+((7+10)×9) 
156 = (2+50)×(107)  157 = (7×25)(2×9)  158 = 2×((7×10)+9)  159 = (2×(25+50))+9 
160 = (7+9)×10  161 = 7×(252)  162 = 9×(257)  163 = (2×50)+(7×9) 
164 = 2+(9×(257))  165 = (7×25)10  166 = (7×25)9  167 = 2+((7×25)10) 
168 = 2+((7×25)9)  169 = (2×(7+(9×10)))25  170 = (72)×(9+25)  171 = (2+7)×(9+10) 
172 = (9×(257))+10  173 = (7×25)2  174 = ((7×25)+9)10  175 = 7×25 
176 = 2×((7×9)+25)  177 = 2+(7×25)  178 = ((7+10)×9)+25  179 = ((2+25)×7)10 
180 = 2×(9×10)  181 = ((2×(7×10))9)+50  182 = 7+((92)×25)  183 = (7×(9+10))+50 
184 = (7×25)+9  185 = (7×25)+10  186 = 2+((7×25)+9)  187 = (2+9)×(7+10) 
188 = (7×(9+25))50  189 = (2+25)×7  190 = 2×((7×10)+25)  191 = ((102)×25)9 
192 = (2+10)×(7+9)  193 = (2×9)+(7×25)  194 = 2×(7+(9×10))  195 = (2×10)+(7×25) 
196 = ((2×9)+10)×7  197 = (9×(252))10  198 = (2+9)×(257)  199 = ((2+25)×7)+10 
You can view the entire results here. (Link opens in a new window). Here is a list of all the sets, like {2,7,9,10,25,50} that can make every single target number from 100999 (Link opens in a new window). 

Of the 1,226 perfect solution sets (sets of numbers that can solve any problem), only five of them contain no large number (for the shortest solution). There is no perfect number set that contains all four large numbers. If you select just one large number, you marginally increase your chances of getting a set of numbers that can solve any problem (cf. selecting two large numbers). The table on the left shows the distribution of large number counts in all the perfect solutions (for the shortest possible solution). 
Pivoting this data the other way, the percentage of problems that are solvable changes with the target number. As the target number gets larger, the percentage of times a random selection of tiles will be able to solve gets smaller. (This is not a surprise. As the target gets larger, it's more likely that the multiplication operator is used more. This will leave gaps in between targets that are not reachable).
All totals below 316 are solvable by >95% of all possible sets.
Other misc pieces of trivia:
The total of 100 is the only total that can be made with one number! (Obviously only if you have the 100 tile).
870 totals are impossible to make without using three or more numbers (whatever numbers are selected).
288 totals are impossible to make without using four or more numbers (whatever numbers are selected).
There are no sets of numbers that allow for all solutions using just five of the numbers. At least six are needed to be sure of being able to solve any problem.
The most 'complicated' total to make is 961. Of the 13,243 possible sets, 5,548 require all six numbers even for their simplest solutions.
The worst tile to draw is a 1. The lowest percentage of solutions occurs when this is in the set.
Out of the 1,226 perfect solution sets, the one with the highest weight (greatest sum) is {2,8,9,50,75,100}, the set with the lowest weight is {2,5,7,8,9,10}. There is only one perfect set with no number less than 8, and that is {8,9,9,10,25,75}.
Try your hand at solving Countdown numbers problems. If you want to find out solutions, I've put a solver online. 
The solver allows you to input the selected tiles (in any order), the requested target, and gives the simplest (smallest number of numbers used) solution to the target number (if an exact solution is possible). If no exact solution is possible the three nearest solutions are given. Have fun!
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