Triangular Art Gallery

Geometry Level 2

How many guards does it take to guard a museum in the shape of a triangle?

Details and Assumptions: A point $P$ in the museum is visible to a guard if the line segment from the guard to $P$ lies within the triangle (or along the boundary).

See Guarding a Museum for more details on guarding a museum.

1 2 3 1 if the triangle is acute, 2 if the triangle is obtuse

6 solutions

Agnishom Chattopadhyay
Mar 12, 2015

The trick is to stand on one vertex and watch the two other vertices

but what about its opposite side?

Rajit Das - 6 years, 3 months ago

That is why I thought the answer was 2. One guarding a vertex and one guarding the other side.

Curtis Clement - 6 years, 3 months ago

Define "guard" as in "placed such that the guard can see the edges and vertices." This is the convention in the classical museum/art gallery problem: $\text{http://en.wikipedia.org/wiki/Art\_gallery\_problem}$ .

Jonathan Yang - 6 years, 3 months ago

This should also tell what it's guarding against. The answer seems if he's standing on top of the wall. What if the theif starts to build a tunnel at the bottom of one of the walls?

Sanjeevan Lakra - 6 years, 3 months ago

true but does that not actually violate the norms......for example...as you say a guard watches two corners from one corner of a triangle....but surely he cant do that simultaneously.......so definitely if he watches one corner quite carefully the other is left unguarded

Arnav Das - 6 years, 3 months ago

But what if the gate is at opposite side of vertex? One guard can guard 2 sides,standing at one corner.

Sachin Arora - 6 years, 3 months ago

the guard can stand anywhere in the triangle

Laura Gao - 3 years, 2 months ago

Skanda Prasad
Mar 13, 2015

Just think in a logical way. Assume that the gallery has only 1 entrance, so only one guard is enough to guard.

true but thieves don't use entrances to enter any museums...right...?

Arnav Das - 6 years, 3 months ago

no, they don't.

Jiya Hansaria - 6 years, 2 months ago

Vikram Venkat
Mar 12, 2015

Maybe, just place them ("him", rather) in the centroid, and he can watch all around by himself. Bit lonely though.

OK but is not that illogical how can a guard stand over a museum and watch all of the sides..... even if its a triangle he could only manage to see the 2 vertices carefully his backside is left off-guard.....for a certain time...

Arnav Das - 6 years, 3 months ago

Lâm Lê
Aug 20, 2020

$\text{You only need 1 guard because you could put him at any point because of the convex walls, he can turn and see any point.}$

Laura Gao
Mar 27, 2018

Convex polygons don't have blind spots. For any two points inside or on the border of a convex shape, there is always a straight line that connects both of them.

Martín Varela
Jan 2, 2018

A single guard at ANY given point inside is enough, because the gallery as a whole is a simple convex polygon (a triangle in this case), which means it cannot have blind spots. (in this ideal mathematical scenario at least)

Triangular Art Gallery

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