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Here are a few fun little puzzles. I haven’t even figured out the third one myself yet.

You must use numbers

1 2 3 4

in any order and each must be used EXACTLY ONCE.

1. You have to find the largest possible number:

i. Using only the operators + - * /

ii. Using only the operators + - * / and powers

AND THEN

2. If you're really bored, find every positive integer that can be formed using + - * / and powers using 1, 2, 3 & 4 exactly once. There are a lot and I'm not sure how many. Who will be first to find them all? :D

(There are no restrictions on the use of operators)

Edited by psychic_mind
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3^4*2+1=163

4*2+1*3=27

No sorry. Neither of those are correct.

Hint: You can use brackets as well. I should have said that in the OP but I thought people may assume that anyway, sorry. :)

Edited by psychic_mind
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No sorry. Neither of those are correct.

Hint: You can use brackets as well. I should have said that in the OP but I thought people may assume that anyway, sorry. :)

Oh yeah I wasn't thinking about brackets. My bad.

(2+1)*(4*3)=36?

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James got part i and White Duck got part ii the first time. You can't string digits together :). Now the hard part remains...

If anyone wants to start off with part (2) then that would be great. Ill post all the ones I can find later.

Edited by psychic_mind
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1 = 1

2 = 2

3 = 3

4 = 4

5 = 4+1

6 = 4+2

7 = 4+3

8 = 4*2

9 = 3^2

10 = 3^2+1

11 = 4*2+3

12 = 4*3

13 = 4*3+1

14 = 4*3+2

15 = 4^2-1

16 = 4^2

17 = 4^2+1

18 = 2*3*(4-1)

19 = 4*(2+3)-1

20 = 4*(2+3)

21 = 3*(4*2-1)

22 = (4*3-1)*2

23 = 4*3*2-1

24 = 2*3*4

25 = 1+2*3*4

26 = (4*3+1)*2

27 = 3^(4-1)

28 = 4*(3*2+1)

29 =

30 = (4+1)*2*3

31 = 2^3*4-1

32 = 2^3*4

33 = 2^3*4+1

Edited by Chicago Ted
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Using Polish notation and a bit of combinatorics you can that there are 15000 different ways to join the number 1,2,3,4 with the operations +,-,*,/,^. Of course we have to rule out the ones that are not integers and the repeated numbers. I think it would be best to make a program that would list them for you.

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__ O __ O __ O __

say __ is for a number, and O for an operator.

Number of ways to fill the __: 4P4

Number of ways to full the O: 5^3

Number of ways to use "("")": 7

Therefore the total ways in which you can form numbers is : 4P4*5^3*7 = 18000

But there are flaw, 1+2+3+4, and 4+2+3+1 are counted twice. Also, some times two sets of operations can lead to the same answer so.

we can correct for the ++ and ** double counts..

_*_*_*_ > 24*7 -1

_+_+_+_> 24*7 -1

_+_+_O_

(_+_)+_O_

(_+_)+(_O_)

_+(_+_O_)

_+_+(_O_)

_O_+_+_

(_O_)+_+_

_O_+(_+_)

(_O_)+(_+_)

(_O_+_)+_

.....................>2*24*5*5-1

similarly for * >2*24*5*5-1

Therefore 18000 - [2*167 + 2*1,199] = 15,268

Only conclusion from all of this:

There should be LESS THAN 15,268 possible numbers.

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For #2: This gets interesting if you start using some of the techniques you could find here. See also Conway chains.

Using "power towers", you could just go on forever, like this: 2^(^(^3(^(^(^(41)))))). I don't know how big this number is. But this number 2^(^4(^(^3))) which, if I read this stuff correctly, I think is also written as 2 => 4 => 3, is equal to = 2^{2^{...^2}} (a tower with 2^16 = 65536 stories). So the short answer is -- there is no answer to #2.

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