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 Forum index » Diversions » Perplex City Puzzle Cards » PXC: Silver Puzzle Cards
[S2] [Dice Set] #230 Sum Other Way
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PuzzledPineapple
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Joined: 07 Apr 2005
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 		[S2] [Dice Set] #230 Sum Other Way
Front:

Quote:
Consider the following equations:

1 = 0 + 12 + 34 + 5 - 67 + 8 + 9
2 = 0 + 123 + 4 -56 -78 +9
3 = 0 + 123 - 45 -6 - 78 + 9
4 = 0 + 12 + 3 - 4 + 5 - 6 - 7 - 8 + 9

The numbers on the left side of each equations are calculated by using each of the digits 0 through 9, in any order, exactly once and either concatenating those digits or separating them with a plus (+) or minus (-) sign.

What is the smallest positive number for which you cannot write such an equation?


Back:

Quote:
Notes found among the papers of Neville Ashberry, antiquary. c. 70-80AC

In a sale of books, purchased three excellent volumes of Varkin, barely foxed, as well as two maps of the City dating from before the cataclysm and a bound copy of the census taken in 15AC, each inscribed with the name Thalian Reese. The last particularly intrigued me, for I have seen very few copies of this document and perhaps none in such fine condition. Upon opening the book, at home, I found some papers held between the pages of the census and, reading them, discovered the pitiable story that I have copied out here. At first, I wondered if it might be a mere fiction but a day's enquiry at the Academy's library confirmed the Reese did indeed experience the loss here documented, that he was, in fact, eventually forced to resign his post, so broken by grief was he.

The name William Oxbow interested me. From his notes in the margins of the census, Rees appears to have tracked down and spoken to anyone with the words Ox or Bow in their name, but with no success. I, however, remembered that it is an old family name of Peterfitch - rather remote in those times and all the more so after the cataclysm. I have written to several members of the family, enquiring whether there is a William among them, aged about 80 years. I have not much hope of receiving a positive response. but am sufficiently intrigued as to wish to continue the hunt.

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PostPosted: Fri Mar 02, 2007 1:48 pm
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X9Tim
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Joined: 29 Sep 2006
Posts: 16

 		A few quick thoughts:

1. The wording of the question should make the answer any number of the form xE-∞ (that's meant to be an infinity symbol)
But I think we can assume that the answer is an integer Laughing

2. The maximum possible answer must be less than 9876543210

Not very helpful, but a starting point i hope Smile

PostPosted: Sun Mar 04, 2007 3:00 pm
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Curlytek
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Joined: 30 Jul 2005
Posts: 112
Location: Melbourne, Australia
X9Tim wrote:
A few quick thoughts:

1. The wording of the question should make the answer any number of the form xE-∞ (that's meant to be an infinity symbol)
But I think we can assume that the answer is an integer Laughing

2. The maximum possible answer must be less than 9876543210

Not very helpful, but a starting point i hope Smile


I don't think that it does, because it doesn't condone the use of decimal points or division, therefore doesn't allow the formation of anything but integers.

Should there be a formula to work this out, or is it simply a "try it till it (really) doesn't work" type of problem?

PostPosted: Sun Mar 04, 2007 9:38 pm
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GasparLewis
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Joined: 19 Nov 2005
Posts: 474
Location: vicinty of NYC

 		Well, think about this.
Not an answer, but something notable.

Spoiler (Rollover to View):
You have 9 spaces between the numbers, with three possibilities: +, -, or nothing (null, which I'll call N). So, since none of the places depend on one another, you get 3^9 = 27^3 = 19,863 possibilities.


This gives two things which can help:

One:
Spoiler (Rollover to View):
The answer would have to be 19,864 or less, and considered repeats and negative results, it's likely far, far, far, far less.


Two:
Spoiler (Rollover to View):
you could make a quick program to run through them all possibilites with the syntax:

0 (+/-/N) 1 (+/-/N) 2 (+/-/N) 3 (+/-/N) 4 (+/-/N) 5 (+/-/N) 6 (+/-/N) 7 (+/-/N) 8 (+/-/N) 9

and see what comes up, then list in order of value to find the first gap in the line, or if you really didn't want to work at all, run a cross-check afterwards until a gap is hit. I don't know exactly what the code would entail, but I'm positive you could whip it up pretty quick, knowing what I know about Visual Basic.


The rest, probably in my own time, I could solve, but that's leagues off.

Besides, I'm never gonna own the card anyway. Laughing

EDIT of EDIT: Took first EDIT, made into a new post.

PostPosted: Sun Mar 04, 2007 10:07 pm
Last edited by GasparLewis on Mon Mar 05, 2007 2:07 am; edited 2 times in total
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classyguy86
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Joined: 15 Jul 2006
Posts: 14
Location: Indiana
GasparLewis wrote:
Well, think about this.
Not an answer, but something notable.

Spoiler (Rollover to View):
You have 9 spaces between the numbers, with three possibilities: +, -, or nothing (null, which I'll call N). So, since none of the places depend on one another, you get 3^9 = 27^3 = 19,863 possibilities.




But the card also says that the digits can be placed in any order. This would add a few possibilities, since concatenating digits in a different order, would definitely change the results.

Spoiler (Rollover to View):
It's been forever since I took probability, but I think the amount of different orders for the 10 digits would be:
(10!) = 3,628,800.
This would make the total possibilities for the equation:
3,627,800*19,863 = 72,078,854,400

(Although some of the equations would be the same)


^^That's just what I remember from 4 years ago, so hopefully a probability guy could check over it.

Hopefully I'm wrong, but if not, this problem might have gotten a bit more difficult.

PostPosted: Sun Mar 04, 2007 10:36 pm
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X9Tim
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Joined: 29 Sep 2006
Posts: 16
GasparLewis wrote:
Well, think about this.
Not an answer, but something notable.

Spoiler (Rollover to View):
You have 9 spaces between the numbers, with three possibilities: +, -, or nothing (null, which I'll call N). So, since none of the places depend on one another, you get 3^9 = 27^3 = 19,863 possibilities.




Yes, but you've missed something...

Spoiler (Rollover to View):
The digits can be in any order, so the number of possibilities = 10^(3^9) = a HUGE number!! Confused


PostPosted: Sun Mar 04, 2007 10:43 pm
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GasparLewis
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Joined: 19 Nov 2005
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Location: vicinty of NYC

 		This is what I get for not reading everything intently.
Confused

Ah well, the principles aren't totally off.

Spoiler (Rollover to View):
And yeah, Classy, you had the idea right, but since I gave the wrong number (it's really 19683, not 19863), it's 71,425,670,400 possibilities, with an answer value lower than 9,876,543,210. Since the number positions ARE dependent on one another, there's 10 options, then 9, then 8... or 10 x 9 x 8..., or 10 factorial. Maybe not as happy as my incorrect reading, but still not miserable. 10^19683 would just be downright unfair.

If I were going to guess, I would say in the neighborhood of... 167,875.

Not that I can verify anything, but I have my reasons for that number.


Wink

PostPosted: Mon Mar 05, 2007 2:17 am
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Rand0m
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Joined: 14 Mar 2006
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Location: London
GasparLewis wrote:
If I were going to guess, I would say in the neighborhood of...


Nope, it ain't that, alas.
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PostPosted: Mon Mar 05, 2007 4:38 am
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KingOfWrong
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Joined: 05 May 2006
Posts: 45
Rand0m wrote:
Nope, it ain't that, alas.

It is within an order of magnitude, though Wink

As for the number of possible sums, they took my code about three hours to work through.

I'd have solved it at 6pm on day 1 except I made two mistakes in the problem definition,
Spoiler (Rollover to View):
first assuming that it was 0..9 in order, with the first digit on its own (i.e 0+... or 0-...), with the answer 211; then any ordering of the digits but still separating the first... oops!
Razz

PostPosted: Mon Mar 05, 2007 4:49 am
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GasparLewis
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Joined: 19 Nov 2005
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KingOfWrong wrote:
Rand0m wrote:
Nope, it ain't that, alas.

It is within an order of magnitude, though Wink


Spoiler (Rollover to View):
That's what I meant; of course I didn't think it was within, like, 10 up or down from that. I just figured that the square root of all possibilities minus the square root of the highest possible answer, rounded out, would make for a good shot in the dark, just on probabilistic principle and the brain lint left over from AP Statistics last year. And, apparently, my idea of scope wasn't too far off.


PostPosted: Mon Mar 05, 2007 5:20 pm
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jwo
Kilroy

Joined: 05 Mar 2007
Posts: 2
KingOfWrong wrote:
As for the number of possible sums, they took my code about three hours to work through.


You should be able to get it down to much less than that (not that it really matters once you have the answer).

Coding hint (not the actual answer):

Spoiler (Rollover to View):

I managed to get my code to get to the solution in under 5 minutes by going for random combinations of digits and operators rather than systematic calculation of all combinations.

As the potential solution gets higher and higher, you can weight the random combinations to favour concatenation over +-. This homes in on the answer pretty quickly.


PostPosted: Mon Mar 05, 2007 5:35 pm
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classyguy86
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Joined: 15 Jul 2006
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Location: Indiana

 		I just have a few questions for those of you who have figured this one out. (not looking for the answer... yet Wink )

Spoiler (Rollover to View):

1. Did any of you do this one without using a computer program?

if no:
2. Now that you have the answer, do you think it would be possible to figure it out without a program?

So far, my lowest guess (using only a calculator, because I'm worthless programming) is 113,575, which is very wrong. So either there's something lower (probably) or I'm completely off.


PostPosted: Mon Mar 05, 2007 11:37 pm
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jwo
Kilroy

Joined: 05 Mar 2007
Posts: 2
classyguy86 wrote:
I just have a few questions for those of you who have figured this one out. (not looking for the answer... yet Wink )

Spoiler (Rollover to View):

1. Did any of you do this one without using a computer program?

if no:
2. Now that you have the answer, do you think it would be possible to figure it out without a program?

So far, my lowest guess (using only a calculator, because I'm worthless programming) is 113,575, which is very wrong. So either there's something lower (probably) or I'm completely off.


No answer given below, but responses to your questions and a hint about the order of magnitude of the answer:

Spoiler (Rollover to View):

1. Yes I used a computer.

2. I would have had trouble getting the answer without a computer. However, there may be some mathematical shortcuts that direct you towards the correct answer (my maths isn't good enough to find any though).

113575 can be calculated by the way (120359-6784). You are correct in assuming there is something lower that can't be calculated and you're not completely off.


PostPosted: Tue Mar 06, 2007 1:57 am
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Albedo
Boot

Joined: 18 Feb 2007
Posts: 34

 		This is encouraging - my programming skills have atrophied over the last decade or so and I never got beyond BASIC so my chances of coming up with a program are fairly slim. However I have been trying to narrow it down and your spoiler above supports my theory that:
Spoiler (Rollover to View):
there are just so many possible combinations of two, three and/or four digit numbers in the sum, any number up to 9999 could almost certainly be generated. Conversely once you get up to using a 6,7, 8 or 9 digit number you don't have many digits left over to 'modify' it with so I was guessing the solution was between five and six digits long. There's hope for me yet!


PostPosted: Tue Mar 06, 2007 7:01 am
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mqpippin
Boot

Joined: 27 Jul 2006
Posts: 29
Location: Houston, Tejas

 		 Frustrated, but in Sum Other Way
Hrm.. none of those smilies look like an annoyed spoiled pouting Programmer....

I write code for a living (mind you, I'm trapped using Basic, Cobol, .ksh shells and sometimes I get to write a .bat file) (no, not VISUAL Basic.. "just" basic)
so... it drives me nuts looking at this card and reading the posts that say "I just wrote a little program that went through all the combinations"... because, I can't for the life of me figure out what kind of logic pattern the code should be in.....

Keeping in mind that I am not trapped in BASIC because that's all I've learned, but instead work for a company that should finally wrap up their migration off of the mainframe server that they bought in the 80's at the end of 1st quarter next year, can someone just give me a VERY generic explain of their logic?

Please do not make fun of me because I have to trouble shoot cobol programs written in 1994 that got missed in the Y2K project and are printing the year 486 on some documents, and 3726 on others. ... and has been for 7 years, but this year, they decided it needs to be fixed....

oh carp... i'm whining again! Sorry. It's probably because I tried to solve Shuffled with the Perfect Shuffle for 7 hours only to find I'd stacked MY deck upside down.... and the 3rd joker with a star on it that my BEST Friend added to my deck while I was out to lunch, after I told her about the Shuffled stuff.... that probably didn't help that I hadn't noticed it...
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PostPosted: Sun Mar 18, 2007 8:59 am
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