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Kevin_P86
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icon The Fours / Nines Puzzle (0)  
I imagine at least some of you have heard of this puzzle before. The aim is to combine exactly four fours (or four nines) with mathematical operators to create an expression evaluating exactly to each number from 0 to 100. For example, the expression 4+4-4-4 would evaluate to 0.

I am offering a +10 to the first person to give me all 101 solutions to the Fours Puzzle, and another +10 to the first person to give me all 101 solutions to the Nines Puzzle. Anyone who submits all 101 solutions to either puzzle after the first person will receive +2.

Here are the operators you are allowed to use for the Fours Puzzle:
Addition: 4 + 4 = 8
Subtraction: 4 - 4 = 0
Multiplication: 4 * 4 = 16
Division: 4 / 4 = 1
Exponents: 4 ^ 4 = 256
Square root: √4 = 2
Decimal: 0.4
Concatenation (*): 44 (using 2 of the 4 fours)
Repeating decimal: 0.44444... = 4/9 (using 1 of the 4 fours)
Factorial: 4! = 24
Brackets to force order of operations: (4 + 4) * 4 = 32
And, of course, any combination thereof.

You are allowed to use the same list of operators for the Nines Puzzle, with the exception of the repeating decimal. I am not going to argue with the fact that 0.9999999.... = 1 (and it happens to be a different discussion altogether), but I find it makes the puzzle too easy if you include it, and I can confirm that all 101 solutions exist without the use of this.

Have fun!

(*) Note: You are not allowed to concatenate intermeiate results. For example, you are not allowed to concatenate (4+4) with (4+4) to make 88.

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[Last edited by Kevin_P86 at 02-04-2007 07:22 PM]
02-04-2007 at 06:41 AM
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TripleM
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icon Re: The Fours / Nines Puzzle (0)  
Is computer assistance allowed? ;)

[Last edited by TripleM at 02-04-2007 08:00 AM]
02-04-2007 at 07:59 AM
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Fafnir
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icon Re: The Fours / Nines Puzzle (0)  
I have about seventy done on the Fours - is there any more efficient way of representing an odd number than 4 / 4 or (4 - 4)!? Or any way of representing an odd number greater than one using only two fours?

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02-04-2007 at 12:49 PM
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krammer
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icon Re: The Fours / Nines Puzzle (0)  
How can you use the infinite decimal if you only have 4 fours...?

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02-04-2007 at 12:58 PM
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Fafnir
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icon Re: The Fours / Nines Puzzle (0)  
Can you negate? For example, could you use -sqrt(4) as -2, or would you have to use something like sqrt(4) - 4? Edit: Assuming you can, I'm up to eighty-four... Edit: No I'm not. Why does forty-four take two fours?! :angry

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[Last edited by Fafnir at 02-04-2007 03:20 PM]
02-04-2007 at 02:59 PM
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zex20913
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icon Re: The Fours / Nines Puzzle (0)  
Can we divide by (4-4)? :P

Seriously though, I have through 30, and then the odd numbers start getting tough.

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02-04-2007 at 03:59 PM
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Albert
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icon Re: The Fours / Nines Puzzle (0)  
I would assume that you can negate square roots like that, because it's just combining them with the subtraction operation. The only possible exception to that would be if the -sqrt4 appears at the start of your expression, but it's also possible that you could rearrange it so that it doesn't anymore.

You can do 4/0.444... = 9 to make an odd number with two 4s. I haven't found any others yet though.
Edit: Also sqrt4/0.4 = 5

[Last edited by Albert at 02-04-2007 05:40 PM]
02-04-2007 at 04:01 PM
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Maurog
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icon Re: The Fours / Nines Puzzle (+5)  
4/.444... looks like four 4s to me.

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02-04-2007 at 04:20 PM
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zex20913
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icon Re: The Fours / Nines Puzzle (+1)  
.444... can be represented with one 4, as a repeated decimal.

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[Last edited by zex20913 at 02-04-2007 04:22 PM]
02-04-2007 at 04:22 PM
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Kevin_P86
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icon Re: The Fours / Nines Puzzle (0)  
TripleM wrote:
Is computer assistance allowed?
I can't really stop you from writing a program to solve this (and that's actually how I got the last few answers, and how I can confirm they all have solutions), but I think it would be better if you'd at least try first without.


Yes, the repeating decimal uses only one four. I suppose I should have clarified that (I have edited the first message to include that).

It is not necessary to have a negative sign in the front of any of the expressions, but I suppose if you must I will accept it.

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02-04-2007 at 07:05 PM
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Kevin_P86
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icon Re: The Fours / Nines Puzzle (0)  
So I just thought I'd make it clear: I realized from my first post that it probably sounded like I was offering +5 to anyone who gave me all the answers, when in fact I meant only the first. So I have changed it to what is above: The first to solve it gets +10, and everyone after that gets +2.

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02-04-2007 at 07:24 PM
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Tahnan
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icon Re: The Fours / Nines Puzzle (0)  
TripleM wrote:
Is computer assistance allowed? ;)
Is googling allowed? :-)
02-04-2007 at 08:36 PM
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Kevin_P86
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icon Re: The Fours / Nines Puzzle (0)  
Tahnan wrote:
TripleM wrote:
Is computer assistance allowed? ;)
Is googling allowed? :-)
No. :)

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02-05-2007 at 12:45 AM
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coppro
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icon Re: The Fours / Nines Puzzle (0)  
Should it be posted in this thread?
02-05-2007 at 01:47 AM
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Kevin_P86
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icon Re: The Fours / Nines Puzzle (0)  
Your choice - you can either post it here, or PM me if you prefer.

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02-05-2007 at 03:31 AM
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Casebier
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icon Re: The Fours / Nines Puzzle (+5)  
Woot! My RPN Calculator really helped me with this problem.
The answer is somewhat long, so I'm putting it in secret tags.
Click here to view the secret text
A word of caution to those doing the 9's puzzle... don't type out the answers too quickly, or you'll accidently skip important keystokes, like I did with this puzzle.

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[Last edited by Casebier at 02-06-2007 10:33 PM : Fixed all errors]
02-06-2007 at 06:39 AM
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Fafnir
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icon Re: The Fours / Nines Puzzle (0)  
I'm going to be evil and point out that 31 = 4!+√4/4+√4 doesn't work - 4!+√4/4+√4 = 24 + 2/4 + 2 != 31, and 4!+√(4/4)+√4 = 24 + 1 + 2 != 31.

But 2/0.4 = 5 - ooooohhh. That works. Why couldn't I have thought of that?

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02-06-2007 at 04:17 PM
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Doom
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icon Re: The Fours / Nines Puzzle (0)  
An interesting puzzle, but feels a bit too tedious for my liking.

Also, in Casebier's post,
25 = 4! + (√4 + 4/4) <- proposed answer
25 = 24 + ( 2 + 1 )
25 = 27
This seems to be wrong as well.
02-06-2007 at 04:46 PM
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Kevin_P86
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icon Re: The Fours / Nines Puzzle (0)  
Yes, 31 is not correct, and it is not the only one (even if you ignore the one mis-matched bracket).

Hint: In the case of a possible ambiguity of order of operations, include brackets. If there are missing brackets, the standard order of operations is applied.

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02-06-2007 at 05:08 PM
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Blondbeard
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icon Re: The Fours / Nines Puzzle (0)  
Is there any numbers which are particulary hard to do? I'm at 26 now, and it feels quite trivial, and I'm a lazy person :P
02-06-2007 at 05:25 PM
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Kevin_P86
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icon Re: The Fours / Nines Puzzle (0)  
Blondbeard wrote:
Is there any numbers which are particulary hard to do? I'm at 26 now, and it feels quite trivial, and I'm a lazy person :P
I suppose that depends. Some of the higher numbers, especially the odd ones, I think are quite challenging.

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02-06-2007 at 05:34 PM
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Blondbeard
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icon Re: The Fours / Nines Puzzle (0)  
I'll try those then...
02-06-2007 at 06:45 PM
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Tim
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icon Re: The Fours / Nines Puzzle (0)  
Blondbeard wrote:
Is there any numbers which are particulary hard to do? I'm at 26 now, and it feels quite trivial, and I'm a lazy person :P
Try 69 and 73.
02-06-2007 at 07:09 PM
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Kevin_P86
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icon Re: The Fours / Nines Puzzle (0)  
And I think the 9's is harder in general than the 4's. I was able to get all the 4's myself, but a few of the 9's I was unable to get without my program.

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02-06-2007 at 07:40 PM
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Casebier
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icon Re: The Fours / Nines Puzzle (+5)  
I'm slowly going through all the rest of my numbers, just in case I made any more oversights.
      ###: \/-Mistake   \/-Correct
Fixed #23: + instead of -
Fixed #25: + instead of -
Fixed #26: Mismatched bracket
Fixed #31: /4 instead of /0.4
Fixed #57: + instead of -
Fixed #96: +4!+4!+4! instead of *4+4-4

All double-checked and corrected. :D

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[Last edited by Casebier at 02-06-2007 10:35 PM]
02-06-2007 at 10:01 PM
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Albert
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icon Re: The Fours / Nines Puzzle (0)  
Blondbeard wrote:
Is there any numbers which are particulary hard to do? I'm at 26 now, and it feels quite trivial, and I'm a lazy person :P
I got all of them except 73 and 89.
Very well done to Casebier for his methods for those two.
02-06-2007 at 10:18 PM
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Casebier
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icon Re: The Fours / Nines Puzzle (0)  
Aye, I was lucky on #73... my RPN calculator can handle 15 digits. Hmm... I just checked, and Window's calculator can handle 15 digits too.

I found two things to be helpful when solving this problem:
1. I wrote down the interesting combinations that I found involving two 4s. (1, 3, 5, 9, 36, 44, 54, 60, 96)
2. I tried solving the problems backwards. I would take the target number and perform different functions on it.

Two parts of my background helped out too:
1. I used to do a lot of mapping in Quake, so I'm used to recognizing those higher multiples of 4.
2. I used to be working towards a Computer Science degree. 4^24 = 2^48 = 6 Bytes.

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02-06-2007 at 10:50 PM
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Kevin_P86
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icon Re: The Fours / Nines Puzzle (0)  
Yes, congratulations, Casebier! It looks good now - however, 96 was ok before (factorial comes before addition...).

I have split the +10 between your two posts (+5 each).

And in regards to 73 - you do not need a calculator that can compute 4^24, since you are immediately taking an 8th root.
(4^24)^(1/8) = 4^(24/8) = 4^3 = 64

The 9's puzzle is still open, and I'm still offering +2 to anyone to solve the 4's puzzle (although I obviously will not accept anyone copying Casebier's answers directly - of course, this does not mean they all need to be different. I will use my discretion).

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02-06-2007 at 11:49 PM
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Maurog
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icon Re: The Fours / Nines Puzzle (+5)  
Here you go, rechecked with calculator. The most trouble for me was finding 32 - was trying to get 2^5 for hours until I found the (very simple) answer.

Click here to view the secret text


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02-07-2007 at 03:51 PM
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Casebier
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icon Re: The Fours / Nines Puzzle (0)  
After doing a little research... Try finding 113 with only 4 4's.

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02-07-2007 at 08:29 PM
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