More or Less

phrontister · 2009-12-08 16:41:06

Here's a tricky little cross-number puzzle. Very logical, and very doable.

Last edited by phrontister (2009-12-08 16:41:33)

bobbym · 2009-12-09 02:48:07

Hi phrontister;

I assume that numbers go in these boxes. For example 16 down might be 11, 1 in each box?

phrontister · 2009-12-09 03:14:45

Hi Bobby,

I assume that numbers go in these boxes. For example 16 down might be 11, 1 in each box?

Yes, that's right.

And if, for example, you then solve 18 across to be 321, the 3 would go in r7c5 and the 2 in r7c6. The 1 in r7c7 is already there from when you entered it as the second digit of 16 down.

Same idea as in a standard crossword puzzle.

mathsyperson · 2009-12-09 05:23:28

This puzzle looks interesting! Is there a unique solution?

phrontister · 2009-12-09 12:01:45

Hi mathsyperson,

Yes - the solution is unique.

Also, all answers are positive integers and none contain leading zeros.

bobbym · 2009-12-09 12:13:14

HI phrontister;

Any duplicates say like 1a equaling 3a?

phrontister · 2009-12-09 14:19:05

Hi Bobby,

Any duplicates say like 1a equaling 3a?

Now there's a question I've never considered before about any cross-number puzzles, or been asked!

Sorry, Bobby, but I don't think I should give the answer to your question because it would colour your thinking - and hence your solving strategy, unnecessarily. It would allow you to make certain assumptions and that might spoil the true logic path. All the information needed to solve the puzzle is there.

Finding a starting point should take a fair bit of good sleuthing work, and even then the rest of the puzzle certainly won't just fall into place. It's a good test of logic skills right to the end.

bobbym · 2009-12-09 14:58:48

Hi phrontister;

OK, I'll try with what I have already.

plutoman · 2009-12-09 18:19:28

Hello,

I tackled this puzzle a year or two ago. I successfully completed it but it certainly wasn't a 5-minute job.
Regarding duplicate answers, well I compile cross-number puzzles from time to time and it's definitely something I'd try to avoid, but not at all costs (unlike crosswords where they're a total no-no).
I can't remember off hand whether this one had duplicate answers.

Last edited by plutoman (2009-12-09 18:20:12)

mathsyperson · 2009-12-09 23:11:00

One way of starting this is to be sneaky, using the solution's uniqueness to help.

For example, 10A (less than 9A) isn't referred to by any of the other clues, so there are no constraints at all on its second digit.
Therefore, the only way that the solution could be unique is if 9A is _1 and 10A is _0.
If 9A ended with anything else, then 10A could take multiple values, which can't happen.

I'd imagine the puzzle is solvable without this style of thinking though.

phrontister · 2009-12-09 23:50:52

Hi mathsyperson,

Sneaky or not - I like it! Well spotted!

This puzzle is at least 40 years old, and no one was that sneaky back then, especially not any cross-number puzzle setters - excluding Rhombus, of course!

Yes - "the puzzle is solvable without this style of thinking". I know that because I solved it, and I'm not sneaky enough to have come up with that trick.

But I just wish I'd thought of it first!

Last edited by phrontister (2009-12-09 23:52:42)

bobbym · 2009-12-14 19:10:20

Hi;

phrontister · 2009-12-14 20:34:54

Hi Bobby,

Well done - all correct. Hope you liked it.

bobbym · 2009-12-15 06:14:00

Hi phrontister;

Yes, I did. Failed to solve the bigger problem of getting the computer to do it for me.

phrontister · 2009-12-15 11:30:46

Hi Bobby,

Failed to solve the bigger problem of getting the computer to do it for me.

William Sit, Professor Emeritus of the City University of New York, says this about that:

It is usually not possible to solve a crossnumber puzzle directly by entering the inter-relationships as equations with the answers to clues as unknowns into a computer algebra system such as Mathematica or Maple.
First, the equations are constrained diophantine equations (the solutions are positive integers within a given range). Without a specially designed package, in general, these off the-shelf systems will not be able to provide much additional information other than returning the same set of equations, perhaps with some trivial rearrangements of the variables.
Second, the equations obtained from the clues are but one aspect of the puzzle. The layout of the grid and locations of the crossed cells provide critical information, too. To use this approach, one would have to treat the digit in each blank cell rather than the answer for each clue as an unknown. It is not easy to break a clue, say of multiplicative type y = ax, into clues for the digits of x and y.
Finally, the ﬁnite-matching clues are also difficult to translate into equations.

So you set yourself a tough task there!

bobbym · 2009-12-15 11:47:05

Hi phrontister;

phrontister wrote:

So you set yourself a tough task there!

Don't believe him.

This rant is totally aimed at the good professor, not at you my friend.

Prof William Sit wrote:

It is usually not possible to solve a crossnumber puzzle directly by entering the inter-relationships as equations with the answers to clues as unknowns into a computer algebra system such as Mathematica or Maple.

Another case of an academic bashing Maple or Mathematica. The FindInstance and Reduce Commands are particularly good at diophantine equations. After I solved it manually, I used those 2 commands to do about 85% of the puzzle. My own ineptness kept me from completing the task in this manner.

Prof William Sit wrote:

First, the equations are constrained diophantine equations (the solutions are positive integers within a given range). Without a specially designed package, in general, these off the-shelf systems will not be able to provide much additional information other than returning the same set of equations, perhaps with some trivial rearrangements of the variables.

Trivial ! Incorrect! Incorrect! Incorrect! This is what I am talking about here in points 3) and 5).

http://www.mathisfunforum.com/viewtopic.php?id=13294

The fact that they are constrained integer problems makes them easier to solve by packages. They just weren't constrained enough for me!

To use this approach, one would have to treat the digit in each blank cell rather than the answer for each clue as an unknown.

I didn't investigate this approach too much but even I could see it was a dynamite method. Why does he make it sound like it is difficult to do by machine?

phrontister · 2009-12-15 14:15:07

Ooo - we're off and running, Bobby!

Sorry, but I can't keep up with much of that. I don't know Maple or Mathematica at all (or any other such programs) - other than messing around a bit with the clever Mathematica solution someone gave me to my Joan's telephone number and my YOB puzzle.

Sit's article is quite a few years old, and may well be outdated.

He is very conversant with crossnumber puzzles and has commented on several - including Little Pigley Farm and NOHOW. He even constructed his own - Lucas-Bonaccio Farm, 1998 - which he "presented to the Math Club at City College on April 24, 1998". That puzzle is beyond me (I've tried...and failed). I might try it again sometime, but according to Sit it "requires a combination of logic, number theory, computing and programming skills, trigonometry, trial and error, and of course, some knowledge about farming".

I'm not defending him, of course...just trying to give you some more (worthless?) background info you may not know.

Dr. Rainer Typke has an online solver at http://www.crossnumber.com/. You can choose from a selection of puzzles submitted by members and solve them online, but you need to register before you can enter a puzzle of your own.

Sit's quote is from the article published on Typke's site (I recall seeing it there many years ago), and in that article Sit also comments on Typke's solver program.

I've entered several puzzles on Dr. Typke's site over the years...including More or Less. For anyone constructing such puzzles the solver is a good means of testing for uniqueness, as it will show all solutions.

I bought Dr. Typke's book "Cross Number Puzzles" that contains 30 good puzzles and has some tips on solving strategies for this genre.

Last edited by phrontister (2009-12-15 19:17:54)

bobbym · 2009-12-15 14:35:31

Hi phrontister;

I know his credentials and I have read his article. He is also an axiom developer so know one would mistake the fact that he is more qualified than I.

in general, these off the-shelf systems will not be able to provide much additional information other than returning the same set of equations, perhaps with some trivial rearrangements of the variables.

But here, better qualifications or not, in my opinion he is not correct. The help of packages is not trivial. My feeling is that a CAS can do a large percentage of the problem. A large percentage cannot be called trivial.

I am familiar with Dr. Typke and some of his work. I didn't know about that site until now though.

To restate, using the clues as variables and the cells as variables, coupled with the recent advances in CAS in my opinion should yield a good percentage of the problem.

phrontister · 2009-12-15 17:52:36

Hi Bobby,

I can understand the interest some have in trying to solve one of these puzzles by programming and the use of formulas and functions, as I had that too until I realised it was all too hard for me. It's a great challenge, and there's the ultimate satisfaction if successful , but also the absolute frustration if unsuccessful!

These puzzles generally just need a calculator and some logic to solve, and are aimed at the general populace - but only those who enjoy doing number puzzles, of course! I actually get more pleasure from just taking time out from the computer and sitting down in the lounge room with one of these puzzles. Then (particularly with these older puzzles), it's just me versus the puzzle setter...on equal footing. Pen, paper and calculator.

I often use Excel or BASIC to help with other puzzles...and that's fun too!

bobbym · 2009-12-15 18:06:06

Hi phrontister;

I like working on them by hand too. They are good mental exercise. But I also like the challenge of getting a computer to work on the same problems.. Don't worry about not being able to program it, I have failed many zillions of times along those lines. As I said this one also proved a bit too tough for my math skills alone.

Bob · 2012-06-28 19:09:02

hi phrontister,

OK, I know it's been a long time arriving. I put this puzzle on my long term 'to-do' list and only recently resurrected it. It hasn't really taken me three years. I thought it would be a good warm up for the DOB puzzles.

Bob

phrontister · 2012-06-29 02:24:36

Hi Bob,

Correct! Excellent!

Background: This was my first 'cross-number' puzzle, given to me by a friend who knew of my liking for number puzzles. It had done the rounds in his circle of acquaintances and nobody had solved it...and then it became my turn to have a go.

It took me a little while to get the picture (I was even younger then than I am now and still a bit wet behind the ears), but eventually light dawned and I got there. I liked the puzzle and started looking around for more, and now I've got quite a nice little collection of them. I've posted a few of them here, but they haven't proven to be the hit that I'd anticipated. Oh, well.

Yes...it's most definitely a good warm-up for the DOB puzzles! I'll expect your answers to them in 2015.

Btw, you may not have noticed, or even been affected by this yet, but I've been playing around with the solution check of bobbym's puzzle to get it where I want it. Still not there yet...got another idea in mind, so there's another update on the way. The main puzzle won't change, cos I'm pretty happy with that. So before embarking on the solution check, just download the latest and greatest version of it.

Another btw: the main puzzle isn't just an exercise in trying to work out what on earth I'm on about...part of it contains a neat (ie, tricky for me) mathematical component that I learnt from the person who solved it.

Math Is Fun Forum

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