# Laser Beams in Helsinki Skies

(The first two chapters are just for flavour, you can safely skip to the TL;DR near the end.)

This puzzle in situated Helsinki for a reason: for the city's "200 years as capital" celebration, the massive granite church in Kallio was illuminated for several nights by shining a powerful blue laser from the Helsinki Observatory, located several kilometres away on the other side of the downtown Helsinki. The installation highlighted (quite literally) an interesting intentional design in the city's planning: Along Unioninkatu, there is an unobstructed line of sight between the two prominent hilltops that are home to those two buildings.

Now then, let's imagine that someone, a lucky drunken university student most likely, happened to gain access to the laser, and after shining it at various buildings, birds, and other targets of opportunity, left the laser so that it happened point exactly to east, and illuminated a ship somewhere near the horizon. (For the purposes of this puzzle, let's assume all these things are possible.) The question is: which end of the laser beam is the northernmost one?

TL;DR: If you point a perfectly straight beam of finite length from Helsinki towards the true east, which end of the beam is the northernmost one?

NB: This is supposed to be a 3D geometry and visualisation problem, or a maths problem without any calculations, if you like. So, although I'm a staunch advocate of loopholes and lateral thinking, please don't post such answers to this puzzle. Well, unless they are particularly excellent, of course.

For extra credit: Would the answer change, if the laser was pointed west instead of east? What if the laser was in Sydney?

EDIT: To avoid any confusion, there is absolutely no trickery involved here. Every term is used in its usual meaning, and there aren't any traps. Some examples:

• Northernmost means "having the largest northern latitude". Latitude is measured relative to the equator. The magnetic north pole has nothing to do with this puzzle.
• Any point that is above ground level is considered to have the same longitude and latitude as the unique point on earth's surface directly below it. (A helicopter directly above the Eiffel Tower is just as north as the Eiffel Tower.)
• East means true east, the direction where earth's rotation is taking every (non-axle) point at every time. At every point, "east" is exactly parallel with the latitude line (or rather, the latitude circle) passing through that point. No compass measurements are involved.
• If you want to, you can choose to include an optional vertical angle between -90 and 90 degrees into your definition of east. ("Point your telescope towards east, and 30 degrees up.") This makes it a lot easier to hit the ship at sea level with a laser on a hilltop. You can also choose to not include the vertical angle; it won't change the answer. (You'll need a somewhat larger ship for the flavour story to make sense, though.)
• This is a pure geometry puzzle, so you know the directions and straightnesses more or less magically; there's no need to account for measurement errors, natural phenomena, blocked lines of sight, or anything of that sort.
• If you make an honest effort to answer the question, I'll of course try to help you wherever I think you might need help, so go on ahead even if you're not sure of your answer!

I think one reasonable definition for a "correct answer" might be "the first factually correct answer that would convince even a somewhat sceptical person."

• Are we accounting for atmospheric refraction or other conditions that bend the beam, or is it assumed to be perfectly straight for the purposes of this question? – Guest Feb 7 '18 at 10:29
• It's not mentioned in the TL;DR where the beam ends - do I understand correctly that the other end is at some object near the horizon? – Lolgast Feb 7 '18 at 10:33
• @Guest, perfectly straight. Also, come to think of it, let's have a beam of constant width, to avoid having to specify that we are interested in the position of the center of the beam. @ Lolgast, it won't matter. The puzzle needs a straight beam with two ends to define the endpoints, which is why the laser needs to hit an object. – Bass Feb 7 '18 at 10:38
• The longer the beam, the easier the problem is :) @Lolgast (If I understood correctly) – Saeïdryl Feb 7 '18 at 10:39
• I'll just point out that if the beam doesn't bend due to atmospheric refraction in the same way that light entering your eyes bends, you won't be able to hit a ship on the horizon with the beam. See Bedford Level experiment. – Guest Feb 7 '18 at 12:26

The ship is

southeast

of the observatory.

For the rest of this answer, I'm putting the observer to the east of the ship, so he's looking west, just so we have somewhere (geographically) to put the ship. It doesn't affect the answer.

We can infer the direction of the ship by checking what direction our instrument is reading when we point at something that is actually west of us.

If the observer and the ship are due east/west of each other, then they're sitting on the same latitude, as shown above. But then the laser beam isn't tangential to the ring that represents our latitude; in other words, the instrument isn't reading the direction it would be reading if we he was traveling around the circle, facing forwards.

He'd have to turn to his left for that to happen:

And then his instrument would be reading "west." So we can conclude that the ship is farther

south

than the observer.

• This took me way too long to get. I think it's basically the same as Lolgast's solution, but I'll leave it here in case it helps anyone else. – Guest Feb 8 '18 at 22:01
• I'm quite convinced by this answer. Then again, I agree with the result already, so I'm very much biased. A sceptic would probably want to question whether you have used a map projection where straight "3d" lines in real world always correspond to straight 2d lines on the map. Also, it wouldn't hurt to be explicit about the location of the surface points below the possible beam endpoints, so that there's no question about them forming a straight line in the first place. This is definitely much more convincing than the other answer with the correct result already. – Bass Feb 11 '18 at 23:03
• Having just used your map to convince a sceptic, I must certainly award the tick now :-) – Bass Feb 13 '18 at 5:20

If I understand this correctly...

The end in Helsinki will be more northernly, for both east and west facing lasers. This can be seen by imagining an infinitely long line, exactly tangential to the globe on Helsinki and going east-west. If you were to wrap that around the globe, you would get a full circle, with the part opposite to Helsinki being as far below the equator as Helsinki is above it. If this were done in Sydney, the result would be the exact opposite - The Sydney end would be the farthest south.

I'd add a picture, but currently don't have the methods/time to do so, will do when I get home.

• V guvax gur yvzvg bs gur orna jvyy nccebnpu gur Rdhngbe vs jenccrq nebhaq gur tybor. (rot13) – Saeïdryl Feb 7 '18 at 10:51
• @Saeïdryl Gur jnl V ivrj vg nf na vapyvarq beovg (frr jvxvcrqvn), jvgu gur yngvghqr bs Uryfvaxv nf vapyvangvba. – Lolgast Feb 7 '18 at 11:00
• Gurer ner vasvavgryl znal yvarf gnatragvny gb gur tybor ba Uryfvaxv, juvpu bar qb lbh pubfr? – Florian Bourse Feb 7 '18 at 12:37
• @FlorianBourse Edited in. – Lolgast Feb 7 '18 at 12:38
• Oh I see it now! Thanks, very nice proof. – Florian Bourse Feb 7 '18 at 12:41

Considering that in the TL;DR it is said:

true east

I would say that:

due to the declination between real(geodetic) north and magnetic north, which is about 9° E in Helsinki, the east end of the beam would be a little nearer to the magnetic north with respect to the end in Helsinki, so the first one is the northernmost. Similarly if the beam points west, the end in Helsinki in the northernmost.

In case the laser was in Sydney:

Since the declination is about 12° E the results are the same as in the Helsinki part.

• There's no need to bring any magnets into this puzzle. – Bass Feb 7 '18 at 11:25