### Email from my brother for the Tim Reeveseses Amongst Us

are you?

Consider the sum cross + roads = danger. Replace each letter with a number

from one to 9. You need to allocate all 9 numbers to a letter. Given that

s = 3, can you work out what 6 digit number represents the word 'danger'

Clue, it may help to set out your working as below to see whats going on.

Have fun :-)

c r o s s

r o a d s +

_______________________________________________

d a n g e r

## 5 Comments:

At 5:06 PM, Timothy V Reeves said…

I got a solution of 96233+62513=158746 (hope it’s right!). However, I didn’t come up with proof that this is the only solution.

The general structure of this problem is not trivial. A wide class of problems starts by defining “constraints”: e.g. in this case s=3, use all digits 1 to 9, etc. The constraints then define a “space” of possibilities or solutions that satisfy these constraints. We then can search through the tentative solutions until we find one that works. “Search, reject and select” as I always say.

The activity of searching often encounters branch points, or places where there is a parting of the ways and sometimes crucial decisions must be made about which route at the branch point is best searched first. So your problem “cross+roads=danger” neatly anticipates what I was going to say about it.

It is also worth noting in passing that “Search, reject, and select” is very “Biblical” in as much as a search enquiry on the Bible like “deity+search+find” comes up with quite a few hits.

At 1:16 PM, Timothy V Reeves said…

Dummy Comment to update Comment number

At 8:51 AM, Laura said…

i don't understand :s

there's a way of working out the problem schematicly though without resorting to trial and error

At 3:34 PM, Timothy V Reeves said…

Right then Ben, ..eh ... Laura, let’s have a look at this one. All problems like this involve traversing a branching maze, a kind of maze search. However, as I said above, where there is a parting of the ways crucial decisions must be made about which route at a branch point is best searched first. What you vaguely refer to as a schematic for arriving at a solution doesn’t change this. The schematic is what programmers call an “algorithm” and the latter constitutes a kind of map that helps you make those crucial branch point decisions. A really good algorithm will get you through the maze with a minimum of backtracking, but even the best algorithms are effectively special cases of the search paradigm, though they may have a very a low backtracking coefficient.

For example: I had never done one of these alphabetical addition problems before and hence it is likely that my methods were rather crude compared to, say, Ben’s brother who probably has to hand the best methods, and no doubt a good brain - I engaged in a lot of inefficient back tracking when I attempted to solve it.

There also remains the meta question: what is the schematic (i.e. algorithm) for finding a efficient schematic method to solve a problem like this? We have no efficient algorithms to solve this meta problem and in fact we have a regress here: How do we find the algorithm, to find the algorithm, to find the algorithm to find …. etc. Hence, it is likely that “search, reject and select” will remain a prominent feature of human life even unto eternity! Thank God! To me mystery is a prey to stalk and to sometimes devour as a carnivore devours its quarry, should the Good Lord provide. What would we do if there were no more mysteries on which to feed and to find spiritual and cognitive sustenance?

One man who is really on top of this algorithm thing is Gregory Chaitin. So if you are really interested (and, boy, do your have to be interested!) go to Gregory Chaitin's web page. C8B. (C8B is my smiley representing a goofy egg-head professor wearing bottle bottomed specs. I’m proud of inventing it!). I got into this algorithm business myself because of its relevance to my studies in probability, randomness, programming and, of course, good old quantum non-linearity. Respectively, that’s a blend of Philosophy, Programming and Physics!

However, it might be a good thing to stay clear of all this, because Ben’s blog is now looking just about as boring as my physics blog. So if I were you Ben, stick to the controversial issues, and you’ll continue to get the odd 20 slightly emotive comments on a post!

At 9:35 AM, Laura said…

Actually with that in mind, we studied Generic Algorithms in Sixth Form... you may be interested in it.

As far as I remember, the principle is being able to write an algorithm for almost any task within certain constraints... Things like Internet ands GPS route planners use them to traverse a number of possibilities quickly and efficiently to find the best route, without resorting to dynamic tree construction.

Linky-link:

http://lancet.mit.edu/~mbwall/presentations/IntroToGAs/

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