Discussion: The Fido Puzzle

In these cold days of winter, some friends sit at their computers, trying to find interesting puzzles.

This puzzle seems to fill the bill. The set-up is very simple. But the result is mystifying.

Does anybody know why this works every time?

The sum of the digits in your final 4 digit number add up to a multiple of 9. I'll leave it to someone else to prove why.

Okay, I think I proved it.

Pick a 4 digit number. Call it abcd.

Rearrange the digits. I'll just reverse them. dcba

Subtract. abcd-dcba = X

I claim that the digits in X add up to a multiple of 9 which means that X is divisible by 9.

So X mod 9 = 0

It also means that:

(abcd-dcba) mod 9 = X mod 9 = 0
(a 10^3 + b 10^2 + c 10 + d) - (d 10^3 + c 10^2 + b 10 + a) mod 9 = 0

Using the fact that: 10^3 mod 9 = 10^2 mod 9 = 10 mod 9 =1

(a 1 + b 1 + c 1 + d) - (d 1 +c 1 +b 1 +a) = 0
a + b + c + d - d - c - b - a = 0
0 = 0

So the result is divisible by 9. Which means the tool just needs to add up your 3 digits and subtract this result from the next multiple of 9 to give you the missing number.

Are you still glad you asked?

A brief summary of the two tools that Jeff used:

1. The difference of the 2 numbers is always a multiple of 9.
Two examples of how a single digit works:
If c is in the 1000's place and moved to the 100's place,
1000c - 100c = 900c = 9*100c, a multiple of 9.
If c is in the 100,000's place and moved to the 10's place,
100,000c - 10c = 99,990c = 9*1110c, a multiple of 9.

2. If a number is divisible by 9, its sum of digits is also divisible by 9. Example: Since 7+8+8+5+8 = 36 is a multiple of 9, then 78,858 is a multiple of 9 (as well as 88857; 88758; etc.)

Winter blahs: Which other website or blog would include a discussion of "cortisol" and "modular arithmetic" on the same day?

Are you still glad you asked?

Man. You guys have no mercy! Modular arithmetic? Is that like modular housing?

I followed jeffw's explanation down to all the "mod 9's" Then he lost me. But that's ok. I never understood the Rule of Nines either.

And I assume all these mod 9's are a fancy way to say that the Rule of Nines applies to this puzzle. However, with the help of the following, I begin to sense a glimmer of understanding with the concept of "wrapping." Am I on the right path?

...Modular arithmetic (sometimes called clock arithmetic) is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value - the modulus. A familiar use of modular arithmetic is its use in the 24-hour clock: the arithmetic of time-keeping in which the day runs from midnight to midnight and is divided into 24 hours, numbered from 0 to 23. If the time is 19:00 now - 7 o'clock in the evening - then 8 hours later it will be 3:00. Usual addition would suggest that the later time should be 19 + 8 = 27, but this is not the answer because clock time "wraps around" at the end of the day...

essentially... except that the answer 27 o'clock would also be correct, just as 19 and 7 refer to the same time in some sense. We usually reset the counter every 12 or 24 hours as only the time (mod 12 or mod 24) is of interest...
However, since Saturdays and Sundays tend to have different schedules, it could be conceivable to only reset the counter once a week (mod 24*7 = mod 168). However people find it hard to do arithmetic with numbers that big, so instead of saying in 79 hours time, it is easier to understand 3 days and 7 hours later.
3 days is the integer part when you divide by 24, 7 is a representative of the "congruency class" of 79 mod 24. If somebody asks you if you will be asleep in 79 hours time, my guess is you will (slap them or) work out that it what time of day it corresponds to - sometimes it is only the congruency class that is useful to consider, hence one interest of modular arithmetic.

question: for which numbers x does the sequence a(n) defined by
a(n) = 2^n x mod 1
satisfy: for each y between zero and one, there exists a subsequence such that the subsequence converges to y ?

Question 2: How big is the set of all such x?

What sense of 'big'? Dense (under the usual metric on R) seems easy to prove, but the complement is also dense.

density is a start, but it isn't necessarily big... countable can be dense but would usually be considered small.
Big tends to come is three varieties:
Hausdorff dimension equal to one. This isn't particularly satisfactory.
A countable intersection of open dense sets: topologically big (or "fat").
Full Lebesgue measure: the metric definition of big - throw a dart at the interval and you are probably guaranteed to hit the set :)

I like 'probably guaranteed' for 'almost surely'.
Indeed I'm aware that 'dense' isn't a particularly impressive form of 'big,' which is why I asked (and you didn't answer) what form of 'big' you wanted.

This thread is a wonderful vignette on the culture of the orienteering community.

The only implication between them is that the third implies the first. I reckon the third is the most interesting. The second is usually used when people can't prove anything stronger. Still, so far there is no way of knowing whether nature prefers the topological or metric version of big. In theory one could come up an experiment...

Even better now you used "vignette" in your post ;p

This thread has nothing to do with orienteering, it's about mathematics. The fact that only orienteers are posting means it is a discussion between 'mathematicians* who orienteer' rather than between 'orienteers who mathematize'. Just like conversations in bars aren't reserved for alcoholics, just for people who happen to have met up in a bar.

* in my case, in the loosest possible sense, in ndobbs' case, in a slightly less loose sense.

http://flickr.com/photos/carlzimmer/sets/721576013...

(math and science tattoos)

I think our participants from the Southern Hemisphere might find a seasonal color scheme biased by 6 months modulo 12.

..The fact that only orienteers are posting means it is a discussion between 'mathematicians* who orienteer'..

I'm beginning to think these are not partial sets, but that all orienteers are closet mathematicians.

This summer, I was at a professional conference, listening to presenters drone on about mumbo-jumbo which didn't interest me very much.

In one presentation, they mentioned a standardized test of "spatial reasoning." As an example, they gave a 3-D perspective drawing of a complicated object, and asked you to mental rotate the object to match another diagram.

This got me wondering...is there a test which might predict if someone has mental abilities conducive to orienteering success? Ideally, it would be context-neutral (ie not specific to maps and/or orienteering).

Since I can, one of these days, I might subject the cadets of the USMAOC to some spatial reasoning tests and see what the correlation is between their test results and their orienteering results.

Of course, a "natural ability" to visualize a map as terrain and terrain as a map isn't going to be a perfect predictor of orienteering success. Training certainly has a large effect.

But, we've all met someone for whom the navigation & mental aspects of our sport come quickly and easily...for them, it just all "makes sense." Others, even muddling along on advanced courses (Green, Red, Blue), always seem to struggle.

Every year, we take 8-12 new cadets into our club, and turn away 15-20 others who come to our try-outs (we are only authorized so many participants...our "orientation to orienteering" drew almost 250 participants, most of whom didn't bother to come to try-outs later). Right now, we base selections on a combination of running fitness, results of some easy navigation courses, and a survey/interview. In my mind, this selection process overweights past experience and a running background. Were we to have a "spatial reasoning" test which might predict future orienteering success, it would seem that ought to factor into our selection process, at least as much as the current criteria.

In an ideal world, we could take everyone onto the team who wanted to try orienteering. As it is, we want to invest our time/effort/money on those who have the most potential.

Anyone know of previous work in this area of predicting orienteering success?

Wow, is USMAOC the only club in North America that actually turns people away? Are cadets allowed to join outside organizations (e.g. HVO, USOF) if they don't make the cut, but still want to orienteer?

and where do you suppose they would find the time?

Frankly, I've never met a good orienteer that didn't take to it immediately. That's not to say that comically huge mistakes aren't made, but anybody who's going to "get" it, get's it right away. I'm not sure how you quantify that, but I would expect that a series of three orange courses would be enough to sort out most samples of interested beginners.

I think the proposed experiment would be fascinating. But you would need more subjects, hence a reason not to turn away all those hopeful applicants.
And this is a pretty good thread morph as well. How did that happen?

How did that happen?

Simplistic puzzle introduced > very first poster provides elegant solution > more complex math discussion befitting the orienteering mind > analysis of appeal of math-centric discussions on A/P > application of these ideas to one current problem: recruitment of new talented orienteers.

I think another common quality is that orienteers are, by nature, problem solvers...we travel the Earth seeking unfamiliar venues, with which we can interact while solving a handful of problems at speed.

"At One with Nature" doesn't come close to describing the appeal of the sport.

"Anyone who is going to get it get it, get's it right away"? Needless to say, that is as fine a selection bias as you are going to see.

Cadets are allowed to join outside organizations (e.g. HVO, USOF) if they don't make the cut. However, most go on to get involved in something else (there are some crazy, 100+ number of clubs available--not all athletic, of course), and we all know the post-graduation USMA orienteering retention rate is pretty low.

Within the 30 members of the club, I always offer free rides and food to attend every local (HVO, WCOC) meet my wife and I attend, and we get but a handful of takers over the course of a year.

So, I don't feel too concerned about the ones we turn away at try-outs. Most years we request to expand the club; most years it gets denied. But, grabbing and working with the most promising candidates at our try-outs doesn't guarantee that we haven't cut the next Eric Bone or Swampfox...then, they turn to some lower sport, like, say, Rugby and we've lost them forever...

Try-outs already consist of a few orange courses. Unfortunately, many seem to commit these "comically huge mistakes" during try-outs. I feel there is a huge entry hurdle to orienteering (basic map-reading, orienteering-specific map symbols, basic skills (handrail, attackpoint, catching feature, etc)) that, in the short term (4 afternoons of try-outs) strongly favors anyone with a bit more experience over a bit more potential.

An example: someone who orienteered for a few weekends with their scout troop, but is not a runner and for whom the navigation doesn't (and won't ever) come easily, is strongly favored over someone who has a natural map-reading talent but is on an O-map for the first (or 2nd or 3rd) time in their life. An answer is more opportunities to familiarize the newcomer before "it counts" and hence level the playing field somewhat.

Unfortunately, our timeline to do this usually looks like 4 or 5 total 2-hour afternoon blocks of time. And, that's why I'm fascinated with the promise of finding a way to screen / test for navigational aptitude that is separate from a direct navigational test or the specifics of orienteering (map symbology, basic skills, etc).

This is intriguing. You could construct a really interesting, cross sectional study. The aptitude test is one thing, but even without it, the effects of experience vs. atheletic aptitude, support systems, retention, etc... and with sucessive cohorts. Very interesting...

Wouldn't you want to measure improvement as well as performance, in that case, at least assuming you get multiple try-outs for the same person? (Without telling people that's what you're doing, so as to avoid incentive problems.)

Re tryouts, have you tried scoring each leg separately instead of total time? Then add the placing positions instead of adding the times. That might diminish the effect of errors and give people 40 opportunities to get something right instead of 3. I wonder how that would affect O-scoring in general. Winsplits comes close to doing it already.

I'm pretty sure I remember an article in an old ONA about spatial reasoning testing of orienteers and fighter pilots but that was studying people already belonging to the two groups rather than trying to predict eventual orienteering ability. Still, there doesn't seem to be any shortage of spatial reasoning tests online. Why not, for starters, pick one (or more) that can be adminstered in a reasonable amount of time and have the current members take it so you can see how well the results correlate with current and, eventually, future ability?

I agree that looking at results in more detail (by looking at splits, or maybe better still at GPS tracks) might tell you a lot more than just the elapsed times. I know I committed some comically huge mistakes early on, and looking at a results list from people trying orienteering for the first time might just be a comparison of how large each persons one large error was, when the interesting stuff is how they did on the rest of the course (which might have taken less time than the one error).

After some thought, I think I also have a slightly simpler explanation if the Fido puzzle (same principle, nothing new here, it's just boiled down some to maybe make it easier to understand to non-mathematicians), as follows:

There's an interesting and well-known property of the number 9 such that if you add up the digits in a number, then add up the digits in the result, and repeat until you have only one digit, then the original number is divisble by 9 if-and-only-if the one digit is 9. And it's also the case that if you get some other digit, that will be the remainder if you divide the number by 9 (which is the same as saying that it's the value of the original number modulo 9).
So, if we take our original number 'x', we could divide it by 9 and get a quotient 'a' and a remainder 'r', so x = 9a+r. If we rearrange the digits to get a new number 'y', we could do the same thing and get y=9b+r. b will be different, but r will be the same, because the number was made out of the same digits, and if we did our little trick of adding digits together, we would have gotten the same answer.
Now we subtract x from y (or y from x), and that's (9a+r)-(9b+r). The 'r' values cancel out, and the result is 9(a-b). And that means that the result is divisible by 9. Which means that the digits will add up to 9, and if we're given all but one of them, we can figure out what the missing one is, because they'll have to add up to a multiple of 9.

I'm pretty sure I remember an article in an old ONA about spatial reasoning testing of orienteers and fighter pilots but that was studying people already belonging to the two groups rather than trying to predict eventual orienteering ability. Still, there doesn't seem to be any shortage of spatial reasoning tests online. Why not, for starters, pick one (or more) that can be adminstered in a reasonable amount of time and have the current members take it so you can see how well the results correlate with current and, eventually, future ability?

An interesting idea, but as one of the participants in the aforementioned study, I'm not so sure it's going to give you any better results than the small series of orange courses. The study was composed of a lengthy series of very brief timed puzzles. In short, for each one we were shown two slides of a simple scene. The second was more or less the first with a rotation applied in one direction or the other. The test was to identify whether it was exactly the same or had been altered so as to not be the same. The slides were shown only for a few seconds each.

My experience was one of an extremely steep learning curve. More of a step function than a curve really. I found the first several trials extremely frustrating because I didn't have enough time to work out the correct solution. Then I suddenly had an a-ha moment and was very easily able to solve all the remaining trials in the alloted time.

When I talked with the woman whose research this was afterward she said this experience was not uncommon. So to a large extent it would seem the results of this particular study were a measure of how quickly you arrived at the a-ha moment where you came up with a short-cut to simplify the mental gymnastics.

Or maybe that's exactly what you want to test potential orienteers for. I wonder what the results would look like if you compared results of a test like this between orienteers and people who've tried O once or a few times and hated it.

Man, I've got a headache!

The second was more or less the first with a rotation applied in one direction or the other. The test was to identify whether it was exactly the same or had been altered so as to not be the same.

Ok it was a long time ago. On further recollection it occurs to me I oversimplified this. Before someone else corrects me, the second scene was the first scene rotated and with an additional object added to the scene. The scenes were of toy houses of various colors and shapes. So each time you looked at a second scene it was rotated from the first in one direction or the other, plus an additional house had been added to confuse you. You essentially had to determine whether the original 2 houses were still in the same orientation to each other, despite the overall rotation of the scene and ignoring the distraction of the added 3rd house.

I finally took a look at this puzzle and am amused that it asks you to write down a number and "make it completely random with lots of different digits." If you are choosing to give it lots of different digits, then it is not completely random :)

Suzanne: What do you expect from a soft-drink commercial??

Besides, not everyone has a father (hi Ricka!) who is a math teacher.