Math (puzzle) question - can you help?

Can anyone help solve this:
A signpost tells you ATHENS is 219 miles away; BUDAPEST is 136; LISBON is 43; and VILNIUS is 74. How far logically is it to HELSINKI?*

Been quite a while since I looked at puzzles like this, so I’m not even sure what sort of things I should be looking out for…


(please add a spoiler box to your answers; thanks)

Is this homework?

Without actually working out the problem, I think I’ll take a stab at giving a hint on how to approach it.

[spoiler]You’re 219 miles from Athens, it says? So get a map, and draw a circle with Athens at the center, having radius of 219 miles. You must be somewhere on that circle.

And you’re 136 miles from Budapest? So draw a circle with Budapest at the center, having radius of 136 miles. You must be somewhere on that circle.

If two circles intersect, they (usually) intersect at two points. Do your two circles intersect at two points? If so, you must be at one of those two points. (If they don’t intersect at all, then either the problem is malformed or you’ve drawn something wrong.)

Can you solve the rest of the problem from here?

(For purposes of this problem, you probably don’t need to worry about taking the curvature of the Earth into consideration, and you can probably use a flat piece of paper for your map. Your answer may be a little distorted or approximate as a result.)[/spoiler]

BTW, it looks like the problem statement is giving you more information than you need to solve the problem. You are given your distance to four different cities. But you should only need to know your distance to three different cities.

You should be able to use any three of the given distance to solve the problem, and then you could use the fourth distance to confirm the result. If you get a contradictory result when you do that, then either you goofed somewhere, or the author of the problem goofed.

The format of the question, its overdetermination, and the fact that Lisbon and Vilnius are actually 2300 miles apart suggests that it’s some sort of word puzzle about the city names rather than anything mathematical or geographical (or that it takes place at something like Epcot). Something to do with summing the letters in each name, noting the position of specific letters, extracting Roman numerals, etc.?

No, that’s not it, Senegoid.

[spoiler]Without even looking at a map I can say with confidence that there is no point which is 43 miles from Lisbon and 74 from Vilnius; those cities are separated by most of the European continent. Nor are any other of the possible pairings geographically feasible.

We’re told that this is a maths puzzle, so we can rule out the possibility of the answer being that there is, say, some state of the US which has its own Athens, Budapest, etc. for which the quoted distances are correct.

My guess is that there is some numerology by which values are assigned to the letters in each name, and those values are combined in some way to produce the associated distances. I think it’s probably significant that the names are all spelled out in upper-case letters, which is not the usual convention.[/spoiler]

Also they are all captial cities, so I excpect the puzzle will also rely on the name of the country the cities belong to. I have no idea though.

Well, as it’s almost 21 years since I was last at school, if it is homework, it’ll be quite a late hand in!

It’s from a rather difficult monthly quiz called the Great Brain Quiz that friends of mine like to test themselves with, and as I have anything but I was seeking help on this puzzle. I don’t actually participate in the quiz myself, but occasionally when everyone is stumped they throw a question or two my way, to see if I can help. And on this occasion I have thrown it onwards to see if you guys could help, as I’ve got nothing.

It’ll be related to vowels / spelling / letter position in alphabet x (?) or letter value etc… I imagine, but 43 being a prime kinda rules out multiplication, so… dunno. Maybe roman numerals in the words minus letter worth…? I was also thinking relative coordinates 21 (up) / 9 (left), or some such, but can’t see anything simple.

The question is in GREAT FUN QUIZ 101

Yep, that’s most likely where it’s from, then. The Great Brain Quiz must be a different monthly quiz that other friends do.

Edit to add - actually, looks like it’s the same quiz, run by the same guy, just renamed / rebadged since the last time I looked.

I can 't imagine a signpost referencing air miles, which would have no practical utility, and the road mileage is far too inexact to be able to solve this problem with any better than about 50% accuracy. The road to Helsinki would need to go through St. Petersburg, and probably through Vilnius, as well. Besides the fact that there is no point within 1.000 miles of both Lisbon and Vilnius, if you are talking about the ones in Portugal and Lithuania.

Furthermore, the fact that the hypothetical sign can be thought to exist at all, implies that it is located in a part of the world where the names of the cities are in English and the metric system is not in place. Absent the above consideration, one cannot “logically” expect a sign to Helsinki to exist at all, nor a distance to Helsinki to be calculable. So the “logical” answer can only be found by setting “logic” aside, which is self-absurd.

My guess: 136
How I arrived at it: I based the mileage on the number of letters in the country..

Furthermore, let’s go back and analyze the language. “A signpost tells you” is not the same thing as saying that the information on the signpost is literally and factually true and reliable. We are certain those are not the actual distances to those actual cities, and therefore, such a hypothetical signpost, offering absolutely no correct information at all, can, logically, be erected at any distance from Helsinki.

I think you are overthinking. These “signpost” puzzles are a standard format. You’re meant to just look at the letters.

I think you must be right, but

in this kind of puzzle there is usually a more satisfying link between the mileage and the letters, rather than just an arbitrary “six letters means A miles, seven letters means B miles” etc. But given the prime number seems to rule out multiplication, and I can’t see any kind of addition working, I guess that is it.

Yeah, I know. Just a shot in the dark. Sorta like my conception.

I would have thought the approach would be more sophisticated than Earl’s suggestion, for instance:[spoiler]

each name contains a number of letters, a number of vowels, a number of “loops” and a number of “curvy” letters (of course, there are many other criteria, the number of strokes required to write a letter for example).

Taking linear combinations of the four suggested variables the mileage is given by the following equation:

1737 x number of letters +
-5675 x number of vowels +
-1323 x number of loops +
2470 x number of curvy letters

So HELSINKI = 1737 x 8 -5675 x 3 -1323 x 0 + 2470 x 1 = -659[/spoiler]

I don’t believe that’s the intended solution, but hey-ho, linear algebra doesn’t lie!

In other words, exactly the same puzzle could have given us the weights of four wrestlers,and we would be asked the weight of a fifth wrestler, and apply the same logic – right? I do not circulate in the society of these “standard formats”. I leave them to the peoplel who use Biblical passages to prove there will be another Hitler in 2033 in Sri Lanka.

OK, an analogous puzzle. Budapest, at the center, has the letters AP, Vilnius has N, etc. AP are the 1st and 16th letters of the alphabet, total 17. The primes above and below 17 are 13 and 19. 17+13+19=49. Add the number of letters in the country (7) and you get 56. Multiply by the number of vowels in the city name (U, A, E), for 118. Subtract the number of adjoining countries that come after the country in Alphabetical order (6) and you get 112 for Budapest. For Helsinki, its 137. Anybody who does not stumble across that arcane numerology fails.

A point to note in the question, it only explicitly has units for Athens. Sometimes for these types of puzzles, they deliberately imply misleading information like that. Is it possible that perhaps the other values represent something else, like angles off a trajectory, lat/long, etc.?