Jump to content
BrainDen.com - Brain Teasers
  • 0


bonanova
 Share

Question

21 answers to this question

Recommended Posts

  • 0

Any constraints? Otherwise...

...should be 1, the smallest non-zero integer...there's a number of ways to do this...

example: rhombus with sides of length 3, and height of 1/3, which gives the small angle to be arctan(1/9).

Edit: corrected my trig

Yes - the constraint is that you don't have a protractor.

Clearly a unit-area 12-perimeter rhombus exists - but you couldn't make it without just guessing the angles.

Edited by bonanova
Link to comment
Share on other sites

  • 0
Any constraints? Otherwise...

...should be 1, the smallest non-zero integer...there's a number of ways to do this...

example: rhombus with sides of length 3, and height of 1/3, which gives the small angle to be arctan(1/9).

Edit: corrected my trig

Yes - the constraint is that you don't have a protractor.

Clearly a unit-area 12-perimeter rhombus exists - but you couldn't make it without just guessing the angles.

Okay. I guess this isn't a lateral-thinking puzzle, but actually I wouldn't necessarily need a protractor...I could do it with a string or a sheet of paper (something I could fold into thirds) or a semi-solid substance (that I could measure out, assuming I know the width of the match) and a clear slide...

Anyways, another question: do all the matches have to be used in the perimeter? Can they be stacked on top of each other?

Link to comment
Share on other sites

  • 0
Okay. I guess this isn't a lateral-thinking puzzle, but actually I wouldn't necessarily need a protractor...I could do it with a string or a sheet of paper (something I could fold into thirds) or a semi-solid substance (that I could measure out, assuming I know the width of the match) and a clear slide...

Anyways, another question: do all the matches have to be used in the perimeter? Can they be stacked on top of each other?

No stacking. End-to-end.

Link to comment
Share on other sites

  • 0
Okay. I guess this isn't a lateral-thinking puzzle, but actually I wouldn't necessarily need a protractor...I could do it with a string or a sheet of paper (something I could fold into thirds) or a semi-solid substance (that I could measure out, assuming I know the width of the match) and a clear slide...

Actually...using no other objects than the matches and pure geometry...

make my rhombus by putting 3 matches end to end horizontally and 1 match vertically at the end, and 1 match vertically at the end of the first match, then use more matches to form a straight line from the beginning of the first match to the vertical match at the end. This line will intersect the vertical match at the end of the first match at exactly 1/3 unit length, and then I can use more matches to draw a straight line perpendicular to the base to mark this 1/3 unit height for my rhombus, then make place my 3 unit side to intersect with this 1/3 unit height line...then remove my marker matches and use them to form the other two sides...

Sorry...but I'm really confused about what the rules are...it seems that we can 'eyeball' 90 degree angles (or else woon's solution wouldn't have worked)...what geometric properties are we allowed to use in conjunction with that?

Link to comment
Share on other sites

  • 0
I'd also argue that woon's solution is three separate polygons.

Yes.

I'm having a hard time ruling out rhombus solutions.

You need to make angles whose sines have rational values - tough to know when you have done that.

But to be fair, you have to eyeball horizontal and vertical to know when a square is exact.

OK. Truce. Let's burn the matches, and state the idea differently. :mellow:

You are given

  1. a sheet of ruled paper, with horizontal and vertical lines 1" apart.
  2. no other notion of distance or angle [don't tear off a piece of the paper to use as a ruler]
  3. an unmarked straight edge
  4. a sharp pencil - which writes only on paper [i.e. won't write on straight edges]
Draw a polygon
  1. whose diameter everywhere is positive definite [ruling out woon's polygons and a 0x6 rectangle]
  2. all of whose sides are integral length
  3. whose perimeter is 12
  4. whose area is integral
What is the minimum area you can achieve?
Link to comment
Share on other sites

  • 0

I come up with 1.

If you make it a 3D cube you use all 12 matches and when seen from the side it only ocupies 1^2 interger. I think this follows the rules set into place.

post-12809-1236259194.jpg

Link to comment
Share on other sites

  • 0

Five. It will look like a short staircase or "W"-ish. I haven't figured out how to space out my matchsticks to draw the diagram. Reducing any further seems to always cause either an overlap of matchsticks or it divides into multiple polygons connected by a common point.

Link to comment
Share on other sites

  • 0

I have a solution with an area of 3 (which I won't give away yet), but I have another question...

[*]a sharp pencil - which writes only on paper [i.e. won't write on straight edges]
Do I have an eraser? Or does everything I write need to be part of the polygon? (i.e., can I cross lines drawn with the straight edge to find a point, then erase the lines?) Or, equivalently, can I use more than one (infinitely long, infinitesimally thin) straight edge? Edited by EventHorizon
Link to comment
Share on other sites

  • 0
I have a solution with an area of 3 (which I won't give away yet), but I have another question...

Do I have an eraser? Or does everything I write need to be part of the polygon? (i.e., can I cross lines drawn with the straight edge to find a point, then erase the lines?) Or, equivalently, can I use more than one (infinitely long, infinitesimally thin) straight edge?

Oops....ignored that the area still needs to be integral. If that was removed and I could use many straight edges, I could get a polygon with as small an area as I want.

My solution of 3 still works, but it's going to be harder to get lower than I previously thought.

Link to comment
Share on other sites

  • 0
Oops....ignored that the area still needs to be integral. If that was removed and I could use many straight edges, I could get a polygon with as small an area as I want.

My solution of 3 still works, but it's going to be harder to get lower than I previously thought.

Alright. Can I hold the straight edge fixed at one point to create a circle?

If so, then the area is 1. Otherwise, I'm stuck at 3 for the moment.

Edited by EventHorizon
Link to comment
Share on other sites

  • 0

Thanks Bonanova for stating the rules clearly...sorry for making you burn the matches...I honestly just wanted to know what the rules were so I could figure out the best solution :(

Anyways...

A 3 sq. unit polygon. It's basically a the 3,4,5 Pythagorean triangle (which has a area of 6 and perimeter of 12), but instead of having a convex right angle, I have 'carved' out of it 3 sq. units.

Trying to depict it as best as possible...7 unit lines arranged as follows (a figure with a total length of 4 and height of 3)


_|
__|
     _|

with a line from the bottom left point to the top right point that is 5 units long.

Edit: Oops...sorry EH if you had the same solution...I was actually thinking about this when I was tossing and turning in bed last night...

Edited by Yoruichi-san
Link to comment
Share on other sites

  • 0
Thanks Bonanova for stating the rules clearly...sorry for making you burn the matches...I honestly just wanted to know what the rules were so I could figure out the best solution :(

Anyways...

A 3 sq. unit polygon. It's basically a the 3,4,5 Pythagorean triangle (which has a area of 6 and perimeter of 12), but instead of having a convex right angle, I have 'carved' out of it 3 sq. units.

Trying to depict it as best as possible...7 unit lines arranged as follows (a figure with a total length of 4 and height of 3)


_|
__|
     _|

with a line from the bottom left point to the top right point that is 5 units long.

Edit: Oops...sorry EH if you had the same solution...I was actually thinking about this when I was tossing and turning in bed last night...

that is not the exact one I had, but it's equivalent.

also uses the 3-4-5 triangle. Start at one vertex, draw right one, then a line of length 5 up 3 and over 4, draw left one, and connect with another line of length 5.

Link to comment
Share on other sites

  • 0

EventHorizon:

Eraser: yes - as if you had matches, you could move some.

Two straight edges: No.

The straight edge is just to align the matches, if we were using matches,

so it's just to draw lines in this case.

Use it to draw a circle: no. the intent is not to construct a figure.

G Threat: Yes it's a polygon but I don't know how you'd guarantee the area is integral

Y-san:

Apology was sweet, but not necessary. -_-

Since I don't smoke or otherwise set fire to things, the matches are no loss.

And I agree, the limits should be clear.

The best puzzles, I've found, have a bit of genius in the wording.

I didn't have the needed genius on this one.

Now to the answers.

EH:

I'd love to see a sketch, cuz I got lost when you mentioned two lines of length 5.

Y-san: One word - Bingo. ;)

Still a bit of a puzzle to word the question and require this solution.

Can't rule out others by saying convex or rectilinear.

I still think this is the only solution that you could really do with matches.

Fairly easy to eyeball vertical 3, horizontal 4, line up the 5, then dent in the corner.

Rhombus seemed qualitatively harder, to me at least.

I read the puzzle a couple days ago and liked it, even tho it didnt come with a solution.

In fact, it asked to make A=4, which I envisioned as 345 with two bottom squares missing.

So You did it one better. I had to draw it to convince myself. Nice.

Link to comment
Share on other sites

Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.

Guest
Answer this question...

×   Pasted as rich text.   Paste as plain text instead

  Only 75 emoji are allowed.

×   Your link has been automatically embedded.   Display as a link instead

×   Your previous content has been restored.   Clear editor

×   You cannot paste images directly. Upload or insert images from URL.

Loading...
 Share

  • Recently Browsing   0 members

    • No registered users viewing this page.
×
×
  • Create New...