Ponder This Challenge - November 2006 - Expressions with 1,2,3,4

Ponder This Challenge:

Puzzle for November 2006.

Consider arithmetic expressions formed from the integers 1,2,3,4 (each of which is to used exactly once) and the operators +,-,* (addition, subtraction, multiplication). We will ask about the values that can be represented by such expressions. For example 1 = (2*3)-(4+1) so 1 can represented. Note 1 = (2+3)-4 would not be a valid representation because it does not use 1.

For this month's puzzle we ask two questions:

1. What is the smallest positive integer that can not be represented by such expressions involving 1,2,3,4.
2. Find a set of 4 distinct positive integers a,b,c,d such that the smallest positive integer that can not be represented by such expressions involving a,b,c,d (instead of 1,2,3,4) is greater than the answer to part 1. You may wish to use a computer for this part.

Revised, 11/3/06, 11:30AM ET: You are allowed to reuse operators (1+2+3+4). You are not allowed to join digits together (12+34). / (divide) is not one of the allowed operators.

We will post the names of those who submit a correct, original solution! If you don't want your name posted then please include such a statement in your submission!

We invite visitors to our website to submit an elegant solution. Send your submission to the ponder@il.ibm.com.

If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Solution

• Click here to view the solution

  Answer:

  1. 29 is the smallest positive integer which can not be represented by expressions involving 1,2,3,4 and the operators +,-,*.
  2. A number of examples exist. The best seem to be {1,2,5,6} and {1,2,5,8} both of which can represent all of the positive integers up to 43.

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  If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Solvers

• Eugene Vasilchenko (11.02.2006 @01:14:48 PM EST)
• (11.02.2006 @01:14:48 PM EST)
• Stephen Steffes (11.02.2006 @01:17:50 PM EST)
• Dave Blackston (11.02.2006 @01:23:42 PM EST)
• V. Balakrishnan (11.02.2006 @01:26:23 PM EST)
• Roberto Tauraso (11.02.2006 @02:27:55 PM EST)
• David Friedman (11.02.2006 @02:54:31 PM EST)
• Mohammed Ajmal (11.02.2006 @03:50:50 PM EST)
• Arthur Breitman (11.02.2006 @03:55:31 PM EST)
• Fýrat Solðun (11/02/2006 04:57:40 PM EST)
• Jan Rubak (11.02.2006 @05:29:34 PM EST)
• Tom Sirgedas (11.02.2006 @05:45:56 PM EST)
• Robin Ryder (11.02.2006 @06:00:44 PM EST)
• Lawrence Hon (11.02.2006 @07:04:21 PM EST)
• Gary M Gerkin (11.02.2006 @08:59:45 PM EST)
• John T. Robinson (11.02.2006 @09:54:14 PM EST)
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• Matthew Samuel (11.02.2006 @11:01:06 PM EST)
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