T
toasty
Ok,
Interesting puzzle, for those who think they are up to it.
I'd be interested to see Empip's solution, he seems a clever chap.
Consider expressions formed from the numbers 1, 2, 3 & 4
(of which each is to used exactly once)
and the symbols +, -, * (addition, subtraction & multiplication).
Example:
To make 1 you could say (2*3)-(4+1) so the number 1 can be represented.
Note: 1 = (2+3)-4 would not be a valid as it does not use the number 1 (you have to use 1, 2, 3 and 4 exactly once)
More examples below:
2 = (2 * 3) - ( 4 * 1)
3 = ( 1 * 2 ) - 3 + 4
4 = 1 + 2 - 3 + 4
5 = 4 + 1 * ( 3 - 2 )
6 = 4 + 3 - 2 + 1
etc...
Question 1) What is the smallest positive number that cannot be represented by such expressions involving 1, 2, 3 & 4?
Note: You are allowed to reuse operators i.e 1+2+3+4 is allowed.
But you are not allowed to join digits together (32+14).
Question 2) Find a another set of 4 positive numbers (to replace 1,2,3 & 4)such that the smallest positive number that cannot be represented is greater than the answer to question 1. i.e a 'better' set of numbers
For example, using 1, 10, 11 and 12 it's impossible to make the number 1. So that would be no good.
Using 3, 4, 6 & 8 you can make all the numbers from 1-11 i.e
1 = 3 - 4 + (8 - 6)
2 = 3 * (4 - 6 +
3 = 4 - 3 - 6 + 8
4 = 4 * (3 + 6 -
5 = 6 - 8 + 4 + 3
6 = (3 * 6) - 4 - 8
etc... etc...
(however using 1,2,3&4 you can do better than this, so that's not the answer either)
Question 3) (for the really interested) What is the ultimate set of 4 numbers? i.e which set of 4 enable you to make the longest series of uninterupted numbers from 1 - ???
For what it's worth, I think I have the correct answer to all 3 questions, but it's taken me hours to find them.. (and lots of CPU cycles!!)
-Dan
Interesting puzzle, for those who think they are up to it.
I'd be interested to see Empip's solution, he seems a clever chap.
Consider expressions formed from the numbers 1, 2, 3 & 4
(of which each is to used exactly once)
and the symbols +, -, * (addition, subtraction & multiplication).
Example:
To make 1 you could say (2*3)-(4+1) so the number 1 can be represented.
Note: 1 = (2+3)-4 would not be a valid as it does not use the number 1 (you have to use 1, 2, 3 and 4 exactly once)
More examples below:
2 = (2 * 3) - ( 4 * 1)
3 = ( 1 * 2 ) - 3 + 4
4 = 1 + 2 - 3 + 4
5 = 4 + 1 * ( 3 - 2 )
6 = 4 + 3 - 2 + 1
etc...
Question 1) What is the smallest positive number that cannot be represented by such expressions involving 1, 2, 3 & 4?
Note: You are allowed to reuse operators i.e 1+2+3+4 is allowed.
But you are not allowed to join digits together (32+14).
Question 2) Find a another set of 4 positive numbers (to replace 1,2,3 & 4)such that the smallest positive number that cannot be represented is greater than the answer to question 1. i.e a 'better' set of numbers
For example, using 1, 10, 11 and 12 it's impossible to make the number 1. So that would be no good.
Using 3, 4, 6 & 8 you can make all the numbers from 1-11 i.e
1 = 3 - 4 + (8 - 6)
2 = 3 * (4 - 6 +
3 = 4 - 3 - 6 + 8
4 = 4 * (3 + 6 -
5 = 6 - 8 + 4 + 3
6 = (3 * 6) - 4 - 8
etc... etc...
(however using 1,2,3&4 you can do better than this, so that's not the answer either)
Question 3) (for the really interested) What is the ultimate set of 4 numbers? i.e which set of 4 enable you to make the longest series of uninterupted numbers from 1 - ???
For what it's worth, I think I have the correct answer to all 3 questions, but it's taken me hours to find them.. (and lots of CPU cycles!!)
-Dan
