# Thinking of a shape

This is on a similar thought process as a 2 year old puzzle on this site: You have one question to tell whether the number I'm thinking of is 1, 2, or 3

I am thinking of a shape.

It is either a triangle or a rectangle or a pentagon.

Ask me 1 question, to which I can only answer Yes, No or Maybe/Sometimes, and guess the shape.

Only one condition: Please dont ask the question based on numbers 3, 4 and 5 which of course correspond to the sides of the respective shapes.

I think there could be multiple answers. Again, I apologize if this has been asked before.

• If the shape in question is irregular, and I ask you a question regarding convex equiangular permutations of your shape, do those questions also apply to the original irregular shape you were thinking of? Apr 7, 2017 at 1:50
• I can do this but first let me tell you about myself! I always like triangles, never squares and sometimes pentagons... So do I like you? Aaaand now I'm off to my lateral-holics anonymous meeting... Apr 7, 2017 at 11:52
• @BrentHackers I'm sorry but I couldn't resist... you're not allowed to ask that question, because although you "sometimes" like pentagons, you don't "sometimes" like that particular pentagon. So the answer to that (if the shape is a pentagon) would be 'I don't know', which is not allowed.
– LS97
Apr 8, 2017 at 9:52
• @LS97 But surely if I sometimes like all pentagons and other times like none, then I'll sometimes like the pentagon in the question? Or at the very least, they could answer 'Maybe' which is allowed! Apr 10, 2017 at 9:30

Do you have an angle above 90 degrees? A triangle is sometimes, a rectangle is no, a pentagon is yes.

• But what about non-Euclidean geometries??! ;) Apr 7, 2017 at 23:57
• You can prove that all pentagons fulfil the condition given fairly easily. Hover this link to see the proof. Apr 8, 2017 at 15:27
• A simple possibility:

Does the shape have four equal angles?

To which the answer would be

YES for the rectangle, NO for the triangle, and SOMETIMES for the pentagon.

• Or if that's too "four"y for you:

Does the shape have three angles summing to 180 degrees?

To which the answer would be

YES for the triangle, NO for the rectangle, and SOMETIMES for the pentagon.

• I think these are problematic because you could well get unreliable answers. Most regular people would probably picture a regular pentagon when you say "pentagon". So they wouldn't necessarily answer "sometimes" or "maybe" for your two questions. Apr 7, 2017 at 14:36
• @PaulLeBeau for the sake of this type of riddle you have to rely of getting correct answers, not what you think a 'regular' person might say. Apr 7, 2017 at 16:26
• @PaulLeBeau What that numbery fellow said. This is a standard thing in logical-deduction puzzles: people are generally assumed to be perfect - and incredibly pedantic - logicians, rather than "regular people". See also. Apr 7, 2017 at 16:28

Perhaps this is a bit broad:

Does the shape have at least two right angles?

For which:

Triangles cannot have two right angles, all rectangles have two right angles, some pentagons have two right angles.

Something that would also work:

Does the shape have at least one pair of parallel sides? [two pairs also works]

For which:

All sides of triangles intersect, all rectangles have two pairs of parallel sides, some pentagons have one or two pairs of parallel sides.

Or:

Does the shape have two pairs of equal sides?

For which:

Triangles don't have 4 sides, all rectangles have two pairs of equal sides, some pentagons have two pairs of equal sides.

Or:

Does at least one of the diagonals pass through the centroid of the shape?

For which:

Triangles don't have diagonals, all rectangles' diagonals pass through their centroid, some pentagons do: Or (a cheeky one):

If I asked you any one of the questions from post 50782, what would be your answer?

For which:

All the questions here have no for the triangle, yes for the rectangle and maybe for the pentagon.

• Doh -- I was trying to be the first to answer, I promise I didn't copy yours :)
– Pat
Apr 6, 2017 at 22:56

The simplest idea i can think of is :

Does the geometric shape you are thinking of has a diagonal and two sides adjacent to it(the diagonal) which follow the Pythagoras theorem?

So, the corresponding answers would be :

Triangle: No diagonals at all.
Rectangle: Yes always.
Pentagon: Sometimes, when one of the angles is a right angle.

• I think it's enough to note that a rectangle has right angles, you don't need to invoke Baudhyan theorem. Apr 7, 2017 at 7:35
• Yes, you're right. Apr 7, 2017 at 7:48

The question would be...

Does your shape have at least 3 right angles?

Because:

A triangle cannot have 3 right angles, so the answer is always "No". A rectangle always has 3 right angles, so the answer is always "Yes". A pentagon can sometimes have 3 right angles, so the answer is "Sometimes"

"Let the division of the sum of the angles of the shape by 3 be $n$. Would it take a hypothetical Champions League knockout 2nd leg $n$ or more minutes to finish (disregarding breaks, penalty shootouts, injury times and the like)?"

Actually,

the answers given to the question linked in the OP will also apply to this question when you tell your partner to divide the sum of the angles by 180 first.

Couldn't you get away with

Is it a rectangle?

Because

If a pentagon has 4 right angles and a 180 degree angle it turns into a rectangle.

• If there is an 180 degree angle, it is no longer a pentagon, because there are only 4 sides. Apr 7, 2017 at 21:00

This problem can be transformed into the original problem (asking about 1, 2, or 3) in the following manner:

Divide the shape you are thinking of into triangles by connecting the shape's vertices with lines that only intersect at vertices of the original shape (meaning your new dividing lines can't cross each other). No such lines will be drawn in a triangle (since it's already a triangle), so the number of contained triangles will be 1. For a rectangle, the number of contained triangles will be 2. For a pentagon, the number of contained triangles will be 3. You can then use any of the valid questions from the original problem (referring to the number of contained triangles) to discover the shape. I would rather not spoil the original problem for anyone who still wants to do it, so I won't actually write the question out here.

Let's acknowledge the fact that:

It may be that this is too related to the number of sides (since there is a direct relation between the number of sides of a polygon and the number of triangles which can be drawn within it in the described manner), but technically I didn't use the numbers 3, 4, and 5, so...I only half-cheated.

Here's a goofy solution for fun:

If I open a standard English dictionary at random to an entry in the section of words that begin with the letter 'T' (excluding acronyms), will the first letter of the name of the shape you are thinking of be the same as one of the first two letters of the word I randomly picked?

Here's the explanation to the goofy solution:

If the shape you are thinking of is a Triangle, your answer will be yes, since the word I picked came from the 'T' section in the dictionary, so every word there will begin with 'T'. If the shape you are thinking of is a Rectangle, your answer will be maybe, because I may or may not have picked a word beginning with 'TR'. If your shape is a Pentagon, your answer would be no, because there is no word in a standard English dictionary beginning with 'TP'.

• What about the free dictionary: thefreedictionary.com/words-that-start-with-tp? What about teepee itself? :D More seriously, what if your rectangle is a square, in which case it would be maybe? Apr 8, 2017 at 10:45
• You'll note that the words all listed there are acronyms. I said excluding acronyms in the post. Also, by definition, all squares are rectangles, and the asker only listed the three shape tyes as possibilities, so they wouldn't be thinking of a square in the first place. Apr 8, 2017 at 17:59