Can You Solve the Four Nines Puzzle?

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Discussion Overview

The discussion revolves around the Four Nines Puzzle, where participants are tasked with using exactly four nines and various mathematical symbols to create expressions that evaluate to every integer between 0 and 132. The conversation includes hints, strategies, and challenges related to solving the puzzle, as well as debates on mathematical concepts such as the value of 0.999... and its equivalence to 1.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • ACG introduces the puzzle and provides an example solution for 0.
  • Some participants discuss the interpretation of .9 repeating as equal to 1, with varying opinions on its validity.
  • Multiple expressions are proposed for various integers, including 9/9+9-9=1 and 68 = 69 - 9/9.
  • Participants express differing views on the use of mathematical symbols, such as square roots and the rotation of the number 9.
  • There is a suggestion that solving higher numbers requires more creativity, with some participants reporting difficulty with specific integers.
  • Discussions arise around the implications of using recurring decimals and their mathematical notation, with some participants questioning the legitimacy of certain expressions.
  • Several participants share their progress and remaining challenges in solving the puzzle, indicating a range of success levels.
  • One participant proposes a related problem involving two 4's and two 9's, suggesting it is more complex.

Areas of Agreement / Disagreement

Participants exhibit a mix of agreement and disagreement, particularly regarding the interpretation of .9 repeating and its equivalence to 1. There is no consensus on the validity of certain mathematical notations and approaches to the puzzle, with multiple competing views remaining throughout the discussion.

Contextual Notes

Some participants express confusion over the notation and implications of recurring decimals, indicating a lack of clarity in definitions and assumptions. The discussion also highlights the subjective nature of creativity in solving the puzzle, with varying interpretations of what constitutes a valid mathematical expression.

ACG
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Hi! I have a puzzle here and was wondering if you guys could solve it.

Using exactly four nines each time and no other numbers or letters, find expressions which evaluate to every integer between 0 and 132. You may put parentheses wherever you wish and uses whatever mathematical symbols you want other than ceiling and floor.

If I remember correctly, the hardest number for me was (I think) 68.

Example:

0 = 99-99.

Good luck!

ACG

P.S.If you want a hint, read on.



















spoiler space.






















_
Hint: .9 = 1.
 
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Hint: .9 = 1.

Err... no it doesn't.
 
ACG is trying to say that .9 repeating = 1, since you can use the repeating symbol as a mathematical symbol.

~Lyuokdea
 
9/9+9-9=1

9/9+9/9=2
 
[itex]68 = 69 - \frac{9}{9}[/itex]
 
The first few are fairly easy. However, the higher the numbers get, the harder they are. By the time you get to 50-60 you have to start to get VERY creative. Rest assured, it can be done.
 
jimmysnyder, that's a smart one ^^
 
but are we allow to rotate the 9? i thoght it woulda changed it into a 6 as it clearly reads six nine not nine nine.

Sqrts allowed?
 
hmmmm... for 2 I thought using .9999 was cheap so i did:


((9*.9)/9)*9
 
  • #10
i don't undertand what does .9=1 or .99999 meant
 
  • #11
ArielGenesis said:
i don't undertand what does .9=1 or .99999 meant

I'm not sure but I think it means:

1 = .9...

so

4 = .9... + .9... + .9... + .9...
 
  • #12
I think they are referring to .9 (with a dot above the 9 to signify a recurring number)

I don't really agree because it will never equal 1 it just will be very close.

In general terms is it cheating to use the shortened form of a square root (e.g. remove the 2)?
 
  • #13
19 down 113 to go :(
 
  • #14
One may not turn the 9's upside down. Sorry about that! :frown:

Square root is legal (and indeed, is crucial). Any mathematical symbol which does not involve letters (sin, cos, etc.) is permitted. Ceiling and floor are prohibited.

The .9=1 thing is

.9 with a bar over it (.9 bar) = .999999... = 1.

ACG
 
  • #15
Here's another hint -- it took me maybe 3 minutes to figure out the solutions for all of the numbers except maybe five or six. Those six took an hour more. Clearly, there's a trick. But what is it?
 
  • #16
3minutes? wow I'm over and hour and I've gotten maybe 3/4s


hour later and I'm down to these #
44,46,65,67,68,74,76,116,
and 116+(i do have a few of these numbers but not a lot)


5min later and I'm left with
65,67,78,74,76,116,118,131 i give up for now
 
Last edited:
  • #17
The .9=1 thing is

.9 with a bar over it (.9 bar) = .999999... = 1.
Is the bar the same as having a dot above the number?

p.s. as I said before, it will still never=1 it will just be really close.

Why would you say it equals 1?
 
  • #18
Proof of 0.999999999... = 1

Let 0.9999999999... = x

Therefore, we have
10x = 9.99999999999...
10x = 9 + 0.9999999999...
10x = 9 + x
9x = 9
x = 1

x = 0.9999999999999... = 1
 
  • #19
:biggrin: Nice bit of trickery :biggrin:
but 0.99999<1 just like 1.000001>1
 
  • #20
Daminc said:
:biggrin: Nice bit of trickery :biggrin:
but 0.99999<1 just like 1.000001>1
what is [itex]1 - .9999... ?[/itex]
 
  • #21
I'm willing to post the solutions on the board if everyone agrees.

In the meantime, I found a second problem.

It's like the Four Nines Problem, except that it's got two 4's and two 9's.

You can get all the numbers from 0 to 232. This is quite a bit harder as there are more options.
 
  • #22
what is 1-.9999
Mmmm, good point :frown:

It's still 0.0001 (with a recuring dot above the zero) which is >0

I agree it's close enough to zero that it makes no practical difference but I still can't agree that it's zero.
 
  • #23
Daminc said:
It's still 0.0001 (with a recuring dot above the zero)
There is no such mathematical notation. The dot over the zero literally means there is no 1, just 0s. Besides, just as a practical matter, I don't see how you can justify that final 1 in terms of the subtraction problem I set you.
 
  • #24
There is no such mathematical notation.
I used to use a recuring dot when I was at school 20 odd years ago.

I never used it in this context but in a way you're dealing with infinities so if you want to have a number that has an infinite number of nines after the decimal then I can have an infinite number of zeros before the 1 :)

Anytime you fix the number of nines then the sum can be calculated.
 
  • #25
Daminc said:
I used to use a recuring dot when I was at school 20 odd years ago.
When I said that there was no such mathematical notation, I was not talking about the dot, I was talking about the 1 at the right hand end after the dot.

Daminc said:
I never used it in this context but in a way you're dealing with infinities so if you want to have a number that has an infinite number of nines after the decimal then I can have an infinite number of zeros before the 1 :)
After good, before bad. I'm allowed to put as many 9s as I want after the decimal point because there is a decimal point, but you are not allowed to put any zeros before the last 1 because there is no last 1. That is what infinity means. in = no, finite = end, infinite = no end = no last digit.

Q: What comes after the last zero?
A: There is no last zero.
Q: What comes before the last one?
A: There is no last one.

I still don't see where that 'last' 1 comes from anyway. Certainly not from the subtraction problem I asked you to perform.
 
  • #26
1-.9 = 0.1
1-.99 = 0.01
1-.999 = 0.001
1-.9999 = 0.0001

etc

for every additional 9 there would be an additional 0
 
  • #27
Daminc said:
1-.9 = 0.1
1-.99 = 0.01
1-.999 = 0.001
1-.9999 = 0.0001

etc

for every additional 9 there would be an additional 0
But 0.0001 is not the final answer. The final anwer is smaller than 0.0001!
 
  • #28
But more than 0 :)
 
  • #29
0.00001 + .99999 = 1.
What is 0.0000...1 + .9999...?
Looks to me like .9999...1.
What do you get?
 
  • #30
Daminc said:
But more than 0 :)
No, smaller than .0000001 as well. In fact smaller than any number that is more than 0.
 

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