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You are standing at the centre of a circular forest of radius 500 metres. The trees of this very regularly planted forest stand in a precise rectangular lattice on the plane, each 10 metres from the next: the points $(m,n)$ within the disc-shaped forest with $m,n\in\mathbb{Z}$ not both zero (since $(0,0)$ is the point where you're standing). Each tree is a perfect cylinder with radius at least 20 centimetres.

Can you see out of the forest?

Source: Leith Hathout, Crimes and Mathdemeanors.

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    $\begingroup$ @tmpearce Because you're also on the same rectangular grid, so there's no two rows you can look directly between. And it's not straightforward :-) $\endgroup$
    – Rand al'Thor
    Commented Sep 13, 2016 at 0:09
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    $\begingroup$ IMO the trees should be given a radius of exactly 20cm. The phrase at least implies it doesnt matter if they are bigger in which case i would say you cant see out. And if you are making them perfectly cylindrical on a perfect grid they might as well be an exact size as well. $\endgroup$
    – gtwebb
    Commented Sep 13, 2016 at 0:10
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    $\begingroup$ To start with, suppose the grid is aligned NS-EW. You only have to consider the angle from North through to North-East (or any of the equivalent sectors between adjacent primary and secondary cardinal directions) because of the eight-fold symmetry. Either it can be done in one and thence all of the sectors, or it can't be done in any of them... $\endgroup$
    – Nij
    Commented Sep 13, 2016 at 0:28
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    $\begingroup$ lateral-thinking solution: yes, just look up :) (although I guess these perfect cyliners extend to infinity :/) $\endgroup$
    – Jonathan Allan
    Commented Sep 13, 2016 at 1:17
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    $\begingroup$ ...hence why it's a comment not an answer $\endgroup$
    – Jonathan Allan
    Commented Sep 13, 2016 at 1:34

5 Answers 5

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Let's call a tree front-row iff there are no other trees directly between you and it. Clearly any non-front-row trees are covered by front-row trees.

We call a pair of front-row trees adjacent iff, from your perspective, no other tree's midpoint appears between the midpoints of those two trees.

Claim: You cannot see through the gap between any two adjacent front-row trees.

If this is true, then clearly you cannot see out of the forest, since any sightline passes between some two adjacent front-row trees.

Proof of claim: Let's take any two adjacent front-row trees $A$ and $B$.

Suppose you can see between them. Then you can see their midpoint $C$, since the line $UC$ (pun intended) has equal distance to $A$ and $B$.

We know that the distance $|UC|$ is greater than $250m$, since otherwise there would be a tree at the integer coordinates $A + B = 2C$ in contradiction to the adjacency of $A$ and $B$.

Now let's look at the area of $UAB$. By Pick's Theorem, we know that $UAB$ has $0.5$ times the area of a grid square, so in this case the area of $UAB$ is $50m^2$.

Now consider the triangles $UCA$ and $UCB$ which make up $UAB$. They each have a base of over $250m$ and a height of at least $20cm$, so they must each be over $25m^2$ large. Together they must therefore have an area of strictly more than $50m^2$, which is clearly a contradiction to the previous paragraph. This proves the claim.

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    $\begingroup$ @randal'thor After submitting this solution, I had a look at the source you gave and oh wow the solution there seems ridiculously overcomplicated. $\endgroup$
    – Anon
    Commented Sep 13, 2016 at 3:23
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    $\begingroup$ @ArturKirkoryan No, it involves proving Minkowski's Theorem using Blichfeldt's Lemma, then using that on an arbitrary 80cm strip across the origin to find a tree whose center lies on that strip (and thus blocks the axis of that strip). Wait a second, that's not true at all, this means that the proof given in the book isn't even correct! And just reducing the strip width to 40cm doesn't work either, because that would break an important inequality necessary for Minkowski's Theorem to work! It appears the proof in the book is in fact irreparably broken. $\endgroup$
    – Anon
    Commented Sep 13, 2016 at 3:53
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    $\begingroup$ @ArturKirkoryan Here you go $\endgroup$
    – Anon
    Commented Sep 13, 2016 at 3:57
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    $\begingroup$ +1 for avoiding Minkowski's theorem. And for "the line $UC$" :) $\endgroup$
    – Rosie F
    Commented Sep 13, 2016 at 5:50
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    $\begingroup$ Brilliant proof! And my sincere apologies for the book proof being wrong; we're lucky that there was still an elementary proof, otherwise my question might have inadvertently been unsolvable or even wrong. $\endgroup$
    – Rand al'Thor
    Commented Sep 13, 2016 at 12:26
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Case 1:

Assumptions: My eyes are in the exact center of my head; The trees are exactly 0.2m in radius.

Result: I cannot see out of the forest. There is a 2mm interference on the tree closest to the edge of the forest blocking my line of sight.

Case 2:

Assumptions: My eyes are like those of a normal human and they are each located 30mm from the center of my head; The trees are exactly 0.2m in radius.

Result: I can see out of the forest. There is a 1.438m gap (measured at the outer edge of the forest) between the edge of the tree closest to me and the edge of the tree farthest from me.

Kudos to the math proof (especially since I can't do that), but I'm an engineer, so I just drew it and measured it in CAD.

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    $\begingroup$ You're hired! Just one more question... What's the volume of this light bulb? $\endgroup$
    – Mazura
    Commented Sep 14, 2016 at 1:48
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    $\begingroup$ Archimedes can tell you.., $\endgroup$
    – Stewbob
    Commented Sep 14, 2016 at 2:30
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Facts:

  1. I am standing
  2. Trees are not less than 20cm radius

Assumptions:

  1. I am an average human, at least 140cm tall (lo-ball estimate)
  2. My eyes are positioned near the top of my body
  3. The majority of trees have a radius below 70cm

Therefore I can easily see out of the forest over the tops of the trees.

NB: This would have been a comment (I don't feel it really meets the spirit of the question) but I don't have the rep.

NNB: I don't perceive this as lateral thinking, just making reasonable assumptions from the stated question, but happy to be argued down!

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    $\begingroup$ The trees are cylindrical, not spherical. And this isn't a lateral-thinking puzzle. $\endgroup$
    – Rand al'Thor
    Commented Sep 13, 2016 at 15:32
  • $\begingroup$ Ah. My bad. So, assuming a cylindrical tree with a reasonable height (let's say, taller thank an average human), no, you can't. That'll teach me to scan questions $\endgroup$
    – jymbob
    Commented Sep 13, 2016 at 15:34
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    $\begingroup$ I'm happy to defer to the proof offered by @McFry above $\endgroup$
    – jymbob
    Commented Sep 13, 2016 at 15:46
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    $\begingroup$ I don't see how your height is relevant to the radius of the trees $\endgroup$
    – njzk2
    Commented Sep 13, 2016 at 17:08
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    $\begingroup$ Having misread the question (and assumed that all trees were spherical, not cylindrical), I was merely pointing out that one can see over the top of short things while standing. $\endgroup$
    – jymbob
    Commented Sep 14, 2016 at 9:18
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Meta answer:

The fact that you say "at least 20cm" suggests that if there is a solution, it must be "no" (that is, the answer is either "it depends on the size" or "no", so if there is a definitive answer, it must be "no").

Lateral thinking answer:

Yes. Just look directly upwards.

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    $\begingroup$ Both "answers" in your answer seem more like comments - neither is taking the question seriously and solving as is clearly intended: as a mathematical problem. $\endgroup$
    – bobble
    Commented Jul 6, 2021 at 21:23
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I note the description says "each 10 metres from the next"; it does not say "10 metres between centres". And each tree has a radius of "at least 20 centimetres". Which taken together would suggest an interpretation where fatter trees have greater distances between their centres.

It does not say what happens to trees that would intersect the circular boundary. I suggest that any tree that is wholy or partly inside the boundary should conform to the planting lattice, even if centred outside the boundary.

I suggest interpreting "see out of the forest" as simply having a clear line of sight to any point outside the boundary, even if it's the surface of a tree that's partly inside the boundary.

I would then posit a configuration of 4 trees, each 490 metres in diameter, planted on the compass points of the border. Even though the like-sized trees planted on the lattice outside the forest protrude into the forest, there's still outside visibility of around 1°20’ of arc at each of the compass bearings 30°, 60°, 120°, 150°, 210°, 240°, 300° & 330°.

The requirement for a regular lattice suggests there could be two sizes of tree, but the fatter trees would still have to be small enough be 10 metres apart on the diagonals (since otherwise that would be the nearest tree).

Enlarging the compass-point trees to a diameter of 690 metres means that the the "small" trees on the alternate lattice points are only 290 m in diameter, and lie entirely outside the forest, visible at the compass bearings of 45°, 135°, 225°, & 315°.

Conversely, reducing the compass point trees to 290m means that the 10 m gaps to the nearest trees are now nearer to square-on to the observer at the centre, leaving wider views beyond the forest.

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  • $\begingroup$ Someone voted this down and didn't even leave a comment explaining why. All of the answers have had to specify additional assumptions, so I don't see why mine is any worse. I intentionally chose different assumptions to make it interesting; there's no point just re-doing the same answer to the same assumptions. $\endgroup$
    – Martin Kealey
    Commented Jul 15, 2021 at 8:29

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