*  C O U N T D O W N  *         
                                         
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    If you've ever watched Countdown on  
  C4 - which seems to have been on since 
  before Noah had even heard of an Ark - 
  you'll be familiar with the "Numbers   
  Game". For those who have managed to   
  miss it all these years, what happens  
  is:                                    
    Six random numbers - five of them    
  usually between 1 and 10, and the      
  sixth usually 25, 50, 75, or 100 - are 
  chosen, and then a "target" number     
  from 100 to 999 is chosen. Two         
  contestants have thirty seconds to     
  manipulate the six numbers to hit, or  
  get close to, the target, using +,-,*, 
  and / freely, separating and combining 
  the numbers in any way. They don't     
  have to use all the numbers.           
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    The solutions can always be written  
  as a single expression. For example,   
  given the numbers 100 2 5 9 8 4 and    
  the target 710, a solution is          
  (100*8)-(2*5*9). A more obscure        
  solution is (9*4+100+8-2)*5.           
  If you want to try a really tough one, 
  which even the near-infallible Carol   
  Vorderman didn't get on the day, try:  
  numbers: 5 7 9 5 3 10 - target: 821.   
  Answer at the bottom.                  
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    Naturally, we of the 8BS persuasion  
  want to know (don't we?) whether a     
  computer can tackle this sort of       
  problem. That's what this program      
  does. It could churn methodically      
  through every possibility, of which    
  there are several zillion, with a lot  
  of duplication, but it seemed more fun 
  to write a program that would produce  
  totally random expressions, evaluate   
  them, and carry on relentlessly until  
  it hit the target. It works, usually,  
  eventually...                          
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    You'll notice from the examples that 
  brackets are necessary for this sort   
  of thing. It's difficult to create a   
  random expression with random          
  brackets, and not much fun trying to   
  evaluate it either, since it would be  
  a string. It was much easier to invent 
  an odd form of algebraic notation      
  which didn't use brackets, but had the 
  same effect. So here, in addition to   
  +,-,*, and /, there is p, m, t, and d  
  - standing for plus, minus etc. Terms  
  connected by these symbols are treated 
  as though in brackets and are          
  evaluated first.                       
    All symbols, first the letters, and  
  then the signs, are evaluated in       
  sequence from left to right. None have 
  precedence.                            
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    So, to hit a target of 711 with      
  numbers 25 8 4 2 5 9, the program      
  could come up with 8t4m5t2+25*9. This  
  is the same as ((8*4-5)*2+25)*9, which 
  becomes 79*9=711.                      
    The numbers and symbols are held in  
  arrays. A string is created just for   
  the display.                           
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    About a third of the expressions     
  produced randomly by the program give  
  a negative answer. Positive answers    
  are shown in red, and negative in      
  magenta. The negative answers are, in  
  effect, valid solutions. They can      
  always be transformed into positive    
  solutions by adjusting the signs and   
  components.                            
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    It's very rare that a solution uses  
  only three of the numbers, and four is 
  unusual, so the program uses four 20%  
  of the time, five 40%, and six 40%.    
    Division has to produce a whole      
  number result, presumably, and is not  
  very helpful. It is used occasionally, 
  just to show willing, but only in the  
  "d" form, never /.                     
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    The program creates and tests random 
  expressions at a rate of about seven   
  per second on the Beeb. It usually     
  comes up with a solution within a few  
  minutes, but can sometimes take ages.  
  The solutions it finds can be neat, or 
  simple, or clever, or obscure, or just 
  plain silly.                           
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    When running the program, you can    
  either enter six numbers and the       
  target - perhaps from the TV program,  
  or it will generate random numbers and 
  a target to aim at.                    
    The scrolling display is a sample of 
  what's happening. It's set initially   
  at 1 in 5. You can change this during  
  "Pause". This does not affect what     
  appears in the box display.            
    It tings when the "closest so far"   
  is within 10 of the target. It also    
  displays alternatives it comes across  
  to the current "closest".              
    When it finds a solution, it stops.  
  You can Continue, to look for another  
  solution, or start a new run.          
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    The answer to 821: 5t9+3*7p10+5.     
  As a test for the program, I invented  
  a tough problem for it, by working     
  backwards from six numbers in a        
  complex sequence to give a target,     
  which I fondly imagined would be       
  tricky and probably unique. It found   
  two alternative solutions...           
    Given 3 5 6 7 7 10 and a target of   
  759, can you find one, or two, or      
  three (or more) solutions?             
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     Press SPACE to return to Options.   
                                         
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