## Small values of 20

A forum for good logic/math puzzles.

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Kerberos
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### Small values of 20

Under what circumstances is the number 10 greater than the number 20? No operations are performed on either of them.

crazyjimbo
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### Re: Small values of 20

Spoiler:
In a negative base?

schmiggen
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### Re: Small values of 20

Spoiler:
Whenever the base that 10 is written in is more than twice as large as the base that 20 is written in. I think that works.

EDIT:
Spoiler:
Can you have a negative base? That's interesting.. hmm...
Kabann wrote:Aw hell, as far as I'm concerned the world started in late 1967. Everything else is just semantics and busy-work.

quintopia
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### Re: Small values of 20

Spoiler:
Sort of. Negative bases sacrifice uniqueness, but if you don't mind that. . .

10 in base -3 = -3
20 in base -3 = -6

Buttons
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### Re: Small values of 20

schmiggen wrote:EDIT:
Spoiler:
Can you have a negative base? That's interesting.. hmm...
Spoiler:
I don't think so. How do you write three in base -2?

EDIT:
Spoiler:
Oh, oops. -1101 works. Huh. As does 111. Wikipedia is enlightening in this matter! Negative bases are actually kind of useful.

Kerberos
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### Re: Small values of 20

Spoiler:
They are both written in the same base.

And I realize now that a negative base works, but it's not what I was thinking of.

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### Re: Small values of 20

Spoiler:
They could be different units; 10 kg is far larger than 20 g

schmiggen
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### Re: Small values of 20

Spoiler:
I suppose this is just semantics and a rewording of the different-units soln., but you can always say that 1s are at least twice as valuable as 2s, (the characters), and 0s aren't valuable at all. Or something along those lines.

Edit: whoops, didn't need to quote the topic question
Kabann wrote:Aw hell, as far as I'm concerned the world started in late 1967. Everything else is just semantics and busy-work.

Mouffles
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### Re: Small values of 20

Spoiler:
For small values of 20
In the spirit of taking things too far - the 5x5x5x5x5 Rubik's Cube.

phlip
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### Re: Small values of 20

Spoiler:
In a number system that's identical to R except that < and > are swapped.

Code: Select all

`enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};void ┻━┻︵​╰(ಠ_ಠ ⚠) {exit((int)⚠);}`
[he/him/his]

Kerberos
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### Re: Small values of 20

Eh, maybe this wasn't as great a puzzle as I thought. But these are some things that I would never dream of doing without hints in the problem:

- Making the numbers different units
- Making up rules like "1 > 2 in my numbering system"
- Switching left and right

Hints to my solution:
Spoiler:
The base is the key.

Spoiler:
While 10 is greater than 20, 1 is not necessarily greater than 2.

Buttons
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### Re: Small values of 20

Hm.
Spoiler:
Base k where 0 < k < 1/2? That's kind of silly. I liked the negative base answer better.

quintopia
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### Re: Small values of 20

base 1/4? How does that work?

jestingrabbit
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### Re: Small values of 20

quintopia wrote:base 1/4? How does that work?

Like base 4 only backwards.

I'm more than a little tempted to call a 169 on this.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

phlip
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### Re: Small values of 20

Surely 101/4 = 1/4 < 201/4 = 1/2...

Code: Select all

`enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};void ┻━┻︵​╰(ಠ_ಠ ⚠) {exit((int)⚠);}`
[he/him/his]

Strilanc
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### Re: Small values of 20

Spoiler:
(mod 19)
Don't pay attention to this signature, it's contradictory.

schmiggen
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### Re: Small values of 20

Strilanc wrote:
Spoiler:
(mod 19)

*acquires a temporary, new hero*
*pauses, wondering if this counts as performing an operation *
Kabann wrote:Aw hell, as far as I'm concerned the world started in late 1967. Everything else is just semantics and busy-work.

HenryS
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### Re: Small values of 20

Strilanc wrote:
Spoiler:
(mod 19)

Spoiler:
Doesn't work, you lose the ordering if you're working in modulo arithmetic.

JamesCFraser
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### Re: Small values of 20

Spoiler:
Surely if they are in the same base, and the bass is not negative, then 10 is always < 20. Given that for a base:

Code: Select all

`"Tens column" : a^1"Units column" : a^010 = 1*a[sup]1[/sup] + 0*a[sup]0[/sup] = a20 = 2*a[sup]1[/sup] + 0*a[sup]0[/sup] = 2aGiven that a is positive, 2 > 1 therfore: 2a > a, Therefore: 20 > 10`

By doing other things, you are merely inventing your own numbering system, surely. In which case, you may as well just define 20 to be < 10. I hope there's a really nice solution to this, lest I should end out feeling cheated.

P.S. I am less than convinced that a negative base makes sense.

Robin S
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### Re: Small values of 20

Well, for example,

Spoiler:
in base -x for x>2
assuming that's the sort of thing you meant.

Edit:
Kerberos wrote:
Spoiler:
I realize now that a negative base works, but it's not what I was thinking of.
Apparently it wasn't.
Last edited by Robin S on Thu Dec 06, 2007 2:59 pm UTC, edited 1 time in total.
This is a placeholder until I think of something more creative to put here.

ptveite
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### Re: Small values of 20

Spoiler:
In golf?

Kerberos
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### Re: Small values of 20

I have been extremely, inexcusably stupid. It turns out that what I thought was the solution is, in fact, not really a solution at all. Sorry for wasting your time.

(Not a) solution:
Spoiler:
I was thinking of base i.

Robin S
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### Re: Small values of 20

Spoiler:
Since in base b 10=b and 20=2b we are looking for b such that b>2b, which as far as I can tell means that 0>b, so it must be a negative base.
Apparently I'm missing something...

Edit: in response to Kerberos' most recent post, that's a bit of relief. I had thought I must be a lot worse with inequalities than I had suspected! For those who have looked at his incorrect solution and aren't sure why it doesn't work, it's basically because
Spoiler:
in base i, 10 and 20 (in other words, i and 2i) are imaginary and hence "greater than" is not meaningful when comparing them. 100 > 200 would be true, however.
Kerberos, you might want to edit the post above which said that
Spoiler:
negative bases weren't what you were thinking of, since they are the only correct solution.
This is a placeholder until I think of something more creative to put here.

Kerberos
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### Re: Small values of 20

Yeah, I was thinking of 100 and 200, not 10 and 20. Henry's post made me realize that I was off by an order of magnitude.

jestingrabbit
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### Re: Small values of 20

No biggee. We all make mistakes. The good people admit them.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

schmiggen
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### Re: Small values of 20

Robin S wrote:Kerberos, you might want to edit the post above which said that
Spoiler:
negative bases weren't what you were thinking of, since they are the only correct solution.

Spoiler:
Is it not true that 10X > 20Y where X > 2Y ? The problem never specifies that they must be written in the same base, so while (like negative bases) this may not be the solution he was looking for, I think it is also a solution.

EDIT: I may have misinterpreted what you meant to say in that quote, so if I did... erm.. whoops.
Kabann wrote:Aw hell, as far as I'm concerned the world started in late 1967. Everything else is just semantics and busy-work.

quintopia
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### Re: Small values of 20

Spoiler:
Kerberos wrote:They are both written in the same base.

Robin S
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### Re: Small values of 20

schmiggen: quintopia's last post was directed at you.
This is a placeholder until I think of something more creative to put here.

schmiggen
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### Re: Small values of 20

*nods*, and whoops. I'd forgotten he had said anything before giving his hints. Danke
Kabann wrote:Aw hell, as far as I'm concerned the world started in late 1967. Everything else is just semantics and busy-work.

JamesCFraser
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### Re: Small values of 20

Kerberos wrote:I have been extremely, inexcusably stupid. It turns out that what I thought was the solution is, in fact, not really a solution at all. Sorry for wasting your time.

(Not a) solution:
Spoiler:
I was thinking of base i.

Spoiler:
10 and 20 wouldn't exist in base i. In base i, numbers would be expressed as unary coefficients of i.

i.e.
111i = 3i10
(1 + i)i = (i - 1)10
(-11111i)i = 510

Saying 20i is like saying 202

Robin S
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### Re: Small values of 20

JamesCFraser, that's a very good point, which I had thought of myself (honestly!) and then completely forgotten about. Silly me. You would need
Spoiler:
base xi, for x>2, an analogy to the negative case mentioned above (base -x, with x>2).
This is a placeholder until I think of something more creative to put here.

gabriel
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### Re: Small values of 20

Spoiler:
Edit: Oh written in the same base. Hmm

wannabe
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### Re: Small values of 20

ptveite, You are getting no love for your clever and accurate response. Good job. I was thinking something along the lines of..

Spoiler:
from a criminal's perspective, the number of lashes he had to endure. Or maybe number of sex partners (i.e. for some, once you get past about 12 the yuck factor creeps in), or how about number of miles to the nearest gas station when you are running on fumes.
Disclaimer: I am not a scientist.

Witticism
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### Re: Small values of 20

wannabe wrote:ptveite, You are getting no love for your clever and accurate response. Good job. I was thinking something along the lines of..

Spoiler:
from a criminal's perspective, the number of lashes he had to endure. Or maybe number of sex partners (i.e. for some, once you get past about 12 the yuck factor creeps in), or how about number of miles to the nearest gas station when you are running on fumes.

Spoiler:
Exactly what I was thinking. Whenever the number in question refers to something you don't want. 10 > 20 when you're golfing

quintopia
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### Re: Small values of 20

@wannabe, witticism (and ptveite). The problem reads "greater than" not "better than." It implies some objectivity, whereas the examples you gave only make the values "greater" from a particular point of view.

Witticism
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### Re: Small values of 20

quintopia wrote:@wannabe, witticism (and ptveite). The problem reads "greater than" not "better than." It implies some objectivity, whereas the examples you gave only make the values "greater" from a particular point of view.

"from a particular point of view"

I believe that this is in line with the description of the problem:

"under what circumstances"

/2cents

jestingrabbit
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### Re: Small values of 20

quintopia wrote:@wannabe, witticism (and ptveite). The problem reads "greater than" not "better than." It implies some objectivity, whereas the examples you gave only make the values "greater" from a particular point of view.

Posting here implies some sense of humour...
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

Hangar
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### Re: Small values of 20

Spoiler:
Aww, I thought you were going for little-endian.

quintopia
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### Re: Small values of 20

Witticism wrote:Posting here implies some sense of humour...

It also implies the tendency to overanalyze things, simply because it is fun to do so.

Anpheus
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### Re: Small values of 20

Ah crap, I wasn't thinking with portals...

Spoiler:
s/>/<
?
Spoiler:

Code: Select all

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