Maths problem 9999

[Wongo]

Use exactly four 9s and any number of arithmathic or higher level math operations to make numbers from 1 to 20.
Example
1=(9+9)/(9+9)
7=(9+9+sqrt9)/(sqrt9)

Here is a cracker from a few years back. Have fun.:D

[watson]

Can't.......got a headache Wongo.......and I will have until you post the answer

[ss_11000]

1 = (9+9) / (9+9) ..................................18 / 18 = 1
2 = (sqrt9 * sqrt9 + 9) / 9 ................................18/9 = 2
3 = (9+9+9) / 9 .................................27 / 9 = 3
4 = (sqrt9 * 9 +9) /9.....................36/9 = 4
5
6 = sqrt9 ((9+9) / 9)............................ 3 * 18/9 = 6
7 =(9+9+sqrt9)/(sqrt9)............................ 21/3 = 7
8 = (9 squared - 9) / sqrt9 * sqrt9...........................72/9 = 8
9 = (((9*9) / sqrt9) /sqrt9)..........................81 / 3 / 3 = 27/3 = 9
10 =( sqrt9 * 9 + sqrt9) /sqrt9.............................30 / 3 = 10
11 = (9 squared + 9 + 9 ) / 9.....................99/9 = 11
12 = sqrt9 + sqrt9 +sqrt9 +sqrt9...................3+3+3+3 = 12
13
14 = (9 squared + sqrt9) / (sqrt9 + sqrt9 )......................(81 + 3) / 6 =
15 = sqrt 9 * 9 - sqrt 9 - 9...................3*9 -12 = 15
16
17
18 = sqrt 9 * sqrt9 + sqrt9 *sqrt9...................9+9 = 18
19
20

i'll do the rest later:rolleyes::2tsup:. its to late to think lol:cool:

[ss_11000]

1 = (9+9) / (9+9) ..................................18 / 18 = 1
2 = (sqrt9 * sqrt9 + 9) / 9 ................................18/9 = 2
3 = (9+9+9) / 9 .................................27 / 9 = 3
4 = (sqrt9 * 9 +9) /9.....................36/9 = 4
5 = (9+sqrt9+sqrt9) / sqrt9.................15/3 = 5
6 = sqrt9 ((9+9) / 9)............................ 3 * 18/9 = 6
7 =(9+9+sqrt9)/(sqrt9)............................ 21/3 = 7
8 = (9 squared - 9) / sqrt9 * sqrt9...........................72/9 = 8
9 = (((9*9) / sqrt9) /sqrt9)..........................81 / 3 / 3 = 27/3 = 9
10 =( sqrt9 * 9 + sqrt9) /sqrt9.............................30 / 3 = 10
11 = (9 squared + 9 + 9 ) / 9.....................99/9 = 11
12 = sqrt9 + sqrt9 +sqrt9 +sqrt9...................3+3+3+3 = 12
13 = (((Sqrt9 + sqrt 9) squared) +sqrt9) / sqrt9 ...........3+3 sqrd +3 / 3 = 39/3 = 13
14 = (9 squared + sqrt9) / (sqrt9 + sqrt9 )......................(81 + 3) / 6 =
15 = sqrt 9 * 9 - sqrt 9 - 9...................3*9 -12 = 15
16
17 (9+9) - (9/9)......1..........................18-1 = 17
18 = sqrt 9 * sqrt9 + sqrt9 *sqrt9...................9+9 = 18
19
20 =

i'll do the rest later:rolleyes::2tsup:. its to late to think lol:cool:.

geez, some of them ^^^ are overly complicated.

eg. 2

9/9 + 9/9 = 1+1 = 2

no wonder my head hurts lol:duh:

[Wongo]

The "9 squared" bit is a bit dodgy.

What's next 9 thirded, 9 tripled or 9 minus five times four ?? :D

[Pat]

Wongo, please do not peddle this #$%^& as it will give everyone a doody of a headache (Except Sheddie):q

[Wongo]

It's fun. Stirlo is enjoying it.:U

[ss_11000]

The "9 squared" bit is a bit dodgy.

What's next 9 thirded, 9 tripled or 9 minus five times four ?? :D

why???

it works:cool: and thats all that matters:D

[Gingermick]

9-9+9²/9=9

[Wongo]

why???

it works:cool: and thats all that matters:D

No it doesn't grasshopper. 9 squared is not the math operation we want here. It is 9 to the power of 2 (we don't want the 2 here) unless you can find a symbol that does that (for example square root):cool:

See.:D

Wax on wax off....:p

[Wongo]

9-9+9²/9=9


Only 9s and symbols sir.

[Ashore]

1 = (9+9) / (9+9) ..................................18 / 18 = 1
2 = (sqrt9 * sqrt9 + 9) / 9 ................................18/9 = 2
3 = (9+9+9) / 9 .................................27 / 9 = 3
4 = (sqrt9 * 9 +9) /9.....................36/9 = 4
5
6 = sqrt9 ((9+9) / 9)............................ 3 * 18/9 = 6
7 =(9+9+sqrt9)/(sqrt9)............................ 21/3 = 7 9 - (9 + 9 )/9 = 7
8 = (9 squared - 9) / sqrt9 * sqrt9...........................72/9 = 8 [( 9 x 9 )-9 ]/9 = 8
9 = (((9*9) / sqrt9) /sqrt9)..........................81 / 3 / 3 = 27/3 = 9
10 =( sqrt9 * 9 + sqrt9) /sqrt9.............................30 / 3 = 10 [( 9 x 9 ) + 9 ] / 9 = 10
11 = (9 squared + 9 + 9 ) / 9.....................99/9 = 11 9 + (9 + 9 ) / 9 = 11
12 = sqrt9 + sqrt9 +sqrt9 +sqrt9...................3+3+3+3 = 12
13
14 = (9 squared + sqrt9) / (sqrt9 + sqrt9 )......................(81 + 3) / 6 =
15 = sqrt 9 * 9 - sqrt 9 - 9...................3*9 -12 = 15
16
17 (9 + 9 ) - ( 9/9 ) = 17
18 = sqrt 9 * sqrt9 + sqrt9 *sqrt9...................9+9 = 18 [(9+9) /9] x 9 = 18

19 ( 9 + 9 ) + (9 / 9 ) = 19
20

i'll do the rest later:rolleyes::2tsup:. its to late to think lol:cool:
Some limited help :rolleyes:

[Ron Dunn]

Anywhere you used 9-squared, just replace it with (9*9).

[ss_11000]

Anywhere you used 9-squared, just replace it with (9*9).
but then i might use two many 9's.:cool:

i see your point now scott...does seem like cheating lol:D


9 + (9 + 9 ) / 9 = 11...ashore, this one doesnt work:?:cool:

[ss_11000]

1 = (9+9) / (9+9) ..................................18 / 18 = 1
2 = (sqrt9 * sqrt9 + 9) / 9 ................................18/9 = 2
3 = (9+9+9) / 9 .................................27 / 9 = 3
4 = (sqrt9 * 9 +9) /9.....................36/9 = 4
5 = (9+sqrt9+sqrt9) / sqrt9.................15/3 = 5
6 = sqrt9 ((9+9) / 9)............................ 3 * 18/9 = 6
7 =(9+9+sqrt9)/(sqrt9)............................ 21/3 = 7
8 = [( 9 x 9 )-9 ]/9 = 8
9 = (((9*9) / sqrt9) /sqrt9)..........................81 / 3 / 3 = 27/3 = 9
10 =( sqrt9 * 9 + sqrt9) /sqrt9.............................30 / 3 = 10
11 = 9 + sqrt9 - 9/9 9 + (9 + 9 ) / 9 = 11

12 = sqrt9 + sqrt9 +sqrt9 +sqrt9...................3+3+3+3 = 12
13 = 9 + sqrt9 + 9/9
14 =
15 = sqrt 9 * 9 - sqrt 9 - 9...................3*9 -12 = 15
16
17 (9+9) - (9/9)................................18-1 = 17
18 = sqrt 9 * sqrt9 + sqrt9 *sqrt9...................9+9 = 18
19 ( 9 + 9 ) + (9 / 9 ) = 19
20 =

i'll do the rest later:rolleyes::2tsup:. its to late to think lol:cool:.

3 left

[Wongo]

9 + (9 + 9 ) / 9 = 11...ashore, this one doesnt work:?:cool:

Why not?

[Wongo]

Only 4 left guys.

13
14
16
20

[ss_11000]

Why not?

9 + (9 + 9 ) / 9 = 11...

9 + 18 / 9 = 27/9 = 3

:?:?:?

[Wongo]

9 + (9 + 9 ) / 9 = 11

9 + (18) / 9

9 + 2 = 11

What is rule #1 in Arithmetic?

Oh dear...:D

[ss_11000]

9 + (9 + 9 ) / 9 = 11

9 + (18) / 9

9 + 2 = 11

Oh dear...:D
:duh::duh::duh:

forget i sed anything would ya:-:doh:

[Greg Ward]

A late start but.....


9 + 9/9 + Square root of 9 = 13

[Wongo]

Only 3 left

14
16
20

....hard aren't they??

[Ashore]

( 9 + 6 ) - 9/9 = 14
( 9 + 6 ) + 9/9 = 16
( 9 + 9 ) + ( 6/sqrt9 ) = 20

you sod wongo I was trying to do my tax today this was the last thing I needed , had to take one of the grandsons for a walk after tea to look at it from out of the box , I take it upside down 9's are ok :U

[ss_11000]

hey wongy:cool:

is this allowed 14 = 99/9 +sqrt9 ??????????????


there's four nines:D

[Ashore]

If upside down 9's arn't then that is
Good one Stirlo :2tsup:

as would be 99/9 + 9 = 20

still leaves 16 though

[Greg Ward]

If 99 isn't allowed:

(((Square root 9)! x 9) + (Square root 9)!)/Square root 9 = 20

greg

[Greg Ward]

9 + (square root 9)! + 9/9 = 16

greg

[Wongo]

Sorry children

No 99, no 6. Just 9.

Greg, I like your approach.:2tsup: It is different from my way but I like it. You got game mate. 16 works but not 20.


2 left...

14
20

[Greg Ward]

let's see.

(square root9)! = 6
6 x 9 =54

54 + (square root 9)! = 60

60/ (Square root 9) = 20

Why doesn't this work?

Greg

[Greg Ward]

9 + (Square root 9)! - 9/9 = 14


Now Wongo..... I've showed you mine, you've got to show yours.....


Greg

[Wongo]

Sorry Greg, there were too many (((((( and I read it wrong. 20 works now. :2tsup: My bad.:D

Well done


:fireworks:

[Greg Ward]

Still want to see your method Wongo, to see what you do here.

Thanks
Greg

[Wongo]

http://www.woodworkforums.ubeaut.com.au/attachment.php?attachmentid=1288&d=1074572624

[ss_11000]

good work wongy, greg and russel:2tsup:

[Ivan in Oz]

9 + (square root 9)! + 9/9 = 16

greg

Please be Gentle with me......

OK!
9 = 9,
and
9/9 = 1,
but WTH!!!:?
(Sq Rt 9)!.......Isn't that 3, NOT 6????? Please Explain

[Wongo]

n! = n * n-1 * n-2 * n-3 * ....* 1

for example

4! = 4 * 3 * 2 * 1
1! = 1

By definition 0! = 1


9 + (square root 9)! + 9/9

= 9 + 3! + 9/9
= 9 + 6 + 1
=16

[Frank&Earnest]

Interesting. I can do 21 and 22, can you?:D

[Frank&Earnest]

No takers? Actually, once you have got the idea it becomes almost mundane up to 64. 65 is a bit challenging and now I am stuck at 67. Think I will go back to woodworking. Anybody out there more determined than I am? :oo:

[Wongo]

I can do 6561 can you?:rolleyes:

:D

[Frank&Earnest]

I can do 6561 can you?:rolleyes:

:D

Wow, that was tough! It took me all night, but I finally cracked it!:U
I still haven't cracked 67 though... does anybody know whether it is even possible?

[ss_11000]

n! = n * n-1 * n-2 * n-3 * ....* 1

for example

4! = 4 * 3 * 2 * 1
1! = 1

By definition 0! = 1


9 + (square root 9)! + 9/9

= 9 + 3! + 9/9
= 9 + 6 + 1
=16
thanx wongo, was wondering what that meant:2tsup:.

now i will know before every one else when we learn that later this year:D

[Groggy]

Sorry Greg, there were too many (((((( Starting to look like LISP code :cool:

[Wongo]

I still haven't cracked 67 though... does anybody know whether it is even possible?

I tried it but couldn't do it. Sorry.

[Frank&Earnest]

Thanks Wongo, once worked out that each 9 can be modified by symbols to make 1, 3, 6 and 9 I thought that maybe all integers could be made up with these four numbers, or something silly like that, but it appears that this hypothesis was too far fetched.:- So, again, back to woodworking!:U

[Koala-Man]

The "9 squared" bit is a bit dodgy.

What's next 9 thirded, 9 tripled or 9 minus five times four ?? :D


I agree. Squared is the power of 2. That's got a 2 in it.

[Ivan in Oz]

And Square Root is the Power to HALF......One on TWO:?:oo:

[Ivan in Oz]

Sq Root is the Power to 0.5.....Same thing

[Frank&Earnest]

The first post said "Use exactly four 9s and any number of arithmethic or higher level math operations." If the operation can be expressed by symbols without numeric terms it seems to be allowed: log (x), !, etc.

The issue here is whether to allow the symbol for square root because by convention the exponent 2 is omitted, while cubic root, for example, is not allowed because the exponent 3 must be written. The 9th root of nine, however, would have to be allowed, unquestionably, but it uses two of the four 9s allowed and is useless anyway.

We need to ask the third umpire.:D

[Wongo]

:clap3: He got game.

[Wongo]

I have a way to do it but I am not going to.

Consider

ln e = 1,

e/e = 1,

ln pi / ln pi = 1

pi / pi = 1

We can have number 1 without using any 9s. :2tsup:

But I won’t use it because it defects the purpose of this thread.

It is a bit like the stuff lawyers do. Legitimate but very very dodgy. :D

[Wongo]

Here is another one

cos(pi) = -1

cos(pi) * cos(pi) = 1

:D

[Frank&Earnest]

Legitimate but very very dodgy. :D

I beg to disagree.:) It is not a legitimate way because the "operation" is not performed on the four nines, it is performed on other numeric entities, albeit represented by symbols (pi, e, etc.). Have you found how to legitimately get 1 from only one 9?

[Wongo]

Have you found how to legitimately get 1 from only one 9?

No bro.

I have been trying to do exactly that. If I could crack that I could crack 67. No luck so far.:doh:

[Frank&Earnest]

infinite root of 9

[Wongo]

Infinite root of 9 doesn't make it 1.

I think the real solution does exist though. The truth is out there.

[zelk]

infinite root of 9

Almost.

Zelk

[Frank&Earnest]

Infinite root of 9 doesn't make it 1.


yes it does... you could write it as lim (x->infinite) of x root of 9, if you want to be that way...:D

[Wongo]

limit is an approximation.:cool:

What is 9 divided by infinity then? Don't tell me it is 0.


:D

[Frank&Earnest]

I guess you are one of those who still refuse to accept that 0.99999... recurrent is =1, then. You are fighting a rearguard battle, but if that's what lifts your boat...:p I'm not a good enough mathematician to follow you down that path.

Just to amuse me: assuming that you could get 1 from one 9, how do you get 67? It still baffles me.

[Wongo]

Hang on mate!!
0.99999... is not 1. Why is it so hard to understand?



Just to amuse me: assuming that you could get 1 from one 9, how do you get 67? It still baffles me.

I have more than 1 way to do it but it is pointless so I won't show you. That will keep you thinking.... :U

[Wongo]

Anyone wants to explain it? :D

http://www.woodworkforums.com/attachment.php?attachmentid=68999&stc=1&d=1205219254

[ss_11000]

i'll take your word for it:cool:

[Frank&Earnest]

Brilliant! :2tsup:Still illegal IMHO, though, for the same reason as before: the operation is not performed on the nines, in this case the nines only identify a subset of an unlimited, and therefore infinite, set of functions. If that were allowed, any and all numbers can be produced.

If, however, you have a solution that accepts the way I suggested of producing 1, I would still be keen to see it. We would then let the professors fight over it. (If you can still find any who champions your cause...:D) I have to admit that I do not have a great opinion of Adelaide academics, though, you might end up being right...:C

[Wongo]

Well it is as illegal as the factorial. Both are mathematics notations not operators. We accepted factorial so we should accept summation too.

I think it is ok. Trust me.:cool:


I will let you know the other 2 ways when I remember them.:D

[Wongo]

i'll take your word for it:cool:

C’mon grasshopper. That is not the right attitude. :U

You need to work it out.

[Frank&Earnest]

Well it is as illegal as the factorial. Both are mathematics notations not operators. We accepted factorial so we should accept summation too.

I think it is ok. Trust me.:cool:


I will let you know the other 2 ways when I remember them.:D

Hmmm.... As I always say to my daughter, never trust anybody who says "trust me"...:p

I still think that not being anally retentive about notations v. operators is OK (it reinforces my acceptance of lim x) , but does not change my point that you are working on infinite functions, not on the four nines, which denies the intention.

[Wongo]

No worries mate. Take away both the ! notation and E notation. (sorry Greg, your solutions for 1 to 20 were wrong :D)

Re the acceptance of lim x, mate let me say it one more time 0.99999... is not 1.:cool: It's not even close.


:D

[Frank&Earnest]

Re the acceptance of lim x, mate let me say it one more time 0.99999... is not 1.:cool: It's not even close.


:D

Well, tell that to my son's calculus professor... I'm too old for this... :C
You wouldn't still be stuck in Euclidean space, perchance?

[Wongo]

It doesn’t matter if it is for a professor in calculus. 0.999999… is not equal to 1. It is a fact. You can round it up to 1 but it is not 1.

[ss_11000]

C’mon grasshopper. That is not the right attitude. :U

You need to work it out.
yeah sure.:rolleyes:

i'll let you know in a couple of months:2tsup:

[Frank&Earnest]

It doesn’t matter if it is for a professor in calculus. 0.999999… is not equal to 1. It is a fact. You can round it up to 1 but it is not 1.

Do you accept that for any two numbers a and b there is an average number in between c = (a+b)/2?

[Wongo]

No I don't

[Frank&Earnest]

f

[ss_11000]

Anyone wants to explain it? :D

http://www.woodworkforums.com/attachment.php?attachmentid=68999&stc=1&d=1205219254



Quote:
Originally Posted by Wongo http://image-mirror.cyanide.com.au/woodworkforums/images/button2/viewpost.gif (http://www.woodworkforums.com/showthread.php?p=699129#post699129)
C’mon grasshopper. That is not the right attitude. :U

You need to work it out.


yeah sure.:rolleyes:

i'll let you know in a couple of months:2tsup:
__________________
S T I R L O :lbs:



gday wongo, i just came across this thread again.

dunno if i'm right but does that (^^^) mean that the sum of the 3rd through to 6th terms of that series is 68, and minus the 1 it equals 67?:?


btw wongo, i got a headband that looks just like your avatar :D

[Wongo]

dunno if i'm right but does that (^^^) mean that the sum of the 3rd through to 6th terms of that series is 68, and minus the 1 it equals 67?:?

Correct

[echnidna]

so how much sawdust is that in cubic metres?

[ss_11000]

sweet. :cool:

we learnt that stuff last term (sequences and series).

[Wongo]

so how much sawdust is that in cubic metres?

1 cubic metre. :2tsup:


:U

[ajw]

Or would that be 9/9 + (9-9) cubic metres?:)

[Afro Boy]

Sorry to dig up an old thread but I found this clock that some of you math-heads might enjoy.

http://geekswithblogs.net/ram/archive/2008/01/04/118249.aspx

[Wongo]

Thats 9 out of 9. :D

[masoth]

Fascinating clock, Afro Boy.

soth

[Skew ChiDAMN!!]

Thats 9 out of 9. :D

Sorry, I'll only rate it 8/9.

Look at the formula for the number 7. :~ It's out by 0.0111...

[Afro Boy]

Thats 9 out of 9. :D

<groan> :;

[Wongo]

Sorry, I'll only rate it 8/9.

Look at the formula for the number 7. :~ It's out by 0.0111...

Stupid isn't it? I suppose it is good enough for primary school students.:rolleyes:

[Skew ChiDAMN!!]

Then again, it's only an analogue clock.

Who needs accuracy for "it's a quarter to nine?"

[Wongo]

Trouble is these people don't have a clue. They are teaching people the wrong stuff.:~ To make it worse, 7 can easily be written as sqrt(9)! + (9/9).:doh:

[adamtas]

The first one I thought of was 99 - (9*9) = 99 - 81 = 18.
Then I saw there were 6 pages of answers. Back to the gardening....

[czk]

It doesn’t matter if it is for a professor in calculus. 0.999999… is not equal to 1. It is a fact. You can round it up to 1 but it is not 1.

but why not just do this, ceiling is a legitimate operator

[Wongo]

Sorry, I am not with you. :?

[Skew ChiDAMN!!]

http://en.wikipedia.org/wiki/Ceiling_function

'Tis a new one on me, but that's nothing new. :rolleyes:

[czk]

the ceiling of x is the smallest integer greater than or equal to x.

floor is probably more common but both are perfectly valid functions

[rosewood]

.

3 left
It,s things like this that make me think i should keep life simple and clean the shed more often.