# Four Triangles Five Shapes puzzle

I discovered here the Four Triangles Five Shapes puzzle (from Emrehan Halici).

From the $4$ triangles, we can form:

— $3$ different parallelograms

— a square

— an isoceles triangle

The sizes and angles of the triangles are approximatively: 1) top left $a=7.5$cm, $b=7.8$cm, $c=3.2$cm ; $80°$ between $a,c$, $25°$ between $a,b$, $75°$ between $c,b$.

2) top right $a=4$cm, $b=5$cm, $c=3.7$cm ; $80°$ between $a,c$, $\sim 48°$ between $a,b$, $\sim 52°$ between $c,b$.

3) bottom left $a=10.5$cm, $b=8.3$cm, $c=4$cm ; $\sim 47°$ between $a,c$, $20°$ between $a,b$, $\sim 113°$ between $c,b$.

4) bottom right $a=7.5$cm, $b=7.5$cm, $c=2$cm ; $85°$ between $a,c$, $15°$ between $a,b$, $80°$ between $c,b$.

I could get none of them… So do you have a hint for some of them?

Thank you very much!

• I don't think that measuring the triangle directly as they appear in the image will be of much help, since that image is in perspective. Additionally, to make a square, we'd need 4 right angles. Since none of the triangles (as measured) have right angles, and only one pair sums to 90 degrees (75 and 15), it seems we cannot make a square from those triangles. I'm hesitant to say we can't answer this, but I do think we need more accurate information. Perhaps someone smarter than I can reverse the perspective transform? Sep 7, 2016 at 15:52
• If someone has this puzzle at home, they could take more accurate measurements, I think. Sep 7, 2016 at 15:59
• Maybe a good idea would be to leave a comment on smallpuzzlecollection.blogspot.ch/2016/04/… Sep 7, 2016 at 16:38
• (Laterally thinking) The blog only says "make", so it does not specify that the shape has to be formed by only the area of the pieces and be two dimensional and continuous, so shapes could be "made" by the hole in the middle of the pieces; we could use less than all four; we could use the third dimension, and possibly other "tricks" (probably short of cutting the pieces!) The least ambiguous of these, in my opinion, is having a hole in the shape or having a hole be the shape. Sep 7, 2016 at 19:39
• Can the square be formed from the negative space? IE: lay the pieces out so that they form the outline of a square? Sep 7, 2016 at 22:40

Based on the measurements in Jerry Loo's answer, I get the following lengths and angles for each of the original triangles:

Shape  -------Side (cm)--------    ------Angle (degrees)------
a         b        c        A (^bc)   B (^ac)   C (^ab)
1      10.1      9.8      2.6      89.1      76.0      14.9
2      9.9       9.9      5.1      75.1      75.1      29.9
3      12.2      9.9      5.1      104.2     51.9      23.9
4      12.3      9.85     7.25     90.7      53.2      36.1


From which I can make the following shapes (in MS Paint, so proportions aren't entirely correct) -

Square:

Parallelogram 1:

Parallelogram 2:

Parallelogram 3:

Isosceles Triangle:

• Thank you very much! I think I will accept your answer (in a few weeks/months) if no better solution is provided. May 1, 2017 at 19:31

I reviewed Emrehan Halici's 4 piece triangle puzzle on my blog mentioned here. I am not good at such puzzles so it was rather difficult for me and even the one solution which I thought I got turned out to be wrong, when I asked Halici for and saw the solutions to the rest.

All the 4 solutions to the four problems do not contain holes in them, nor do they overlap..they are just placed side by side flat on a surface (no gaps) to form the 5 shapes required, all 2D, no 3D no tricks...seems easy when you look at the solutions with hindsight.

If you want the dimensions and angles, its best you contact Halici directly. I have left a reply to the comment made by one of your members on my blog. Please PM me and I will ask if Halici is willing to allow a direct email to him for specs, info, etc...

At this year's IPP36 this past August in Japan, we exchanged puzzles and his puzzle is again one of these several-pieces-make-shapes type puzzle. I will post a write up on my blog in due course.

This is a very interesting forum/site relating to puzzles and I am learning a lot. Thanks

Jerry

• Welcome to puzzling :) Sep 8, 2016 at 5:48
• This doesn't seem to answer the question. And I am not sure you can promote your site like this. Sep 8, 2016 at 6:27
• @Jerry: Can the pieces be flipped? Sep 8, 2016 at 8:42
• @Jerry: do you remember the solutions? If so, you could take pictures of them… Or at least take a picture of the 4 triangles without any perspective, so that we can measure the angles/sides. Sep 8, 2016 at 9:11
• Hi, thanks for joining up here to post an answer! (Note to anyone who might think this is spam: the website mentioned in this answer is the site where the OP found the puzzle, so it's actually relevant to the question, and the answerer's affiliation is clearly disclosed.) Sep 8, 2016 at 11:26

here are the measurements for each of the pieces. Sorry I can't provide the angles as I don't have any instrument that can measure angles. Hope this helps.

• It's not clear (to me) how this helps solving the puzzle. The OP already provided their own measurements. Apr 27, 2017 at 11:13
• Are you Jerry's Mechanical Puzzles, but using a different account? Because I think multi-accounting is against the site rules. Apr 27, 2017 at 13:14
• Dear Jerry, can the pieces be flipped to solve the puzzle? And how many of the 4 triangles have a right angle (90°)? Apr 28, 2017 at 16:07
• By the way, I would be glad to have some solution(s) to these puzzles! Apr 28, 2017 at 16:07
• @Glorfindel: The OP appears to have derived the measurements in the question by printing out the image, then measuring the shapes on the printout - this would only produce accurate proportions if the image had been shot directly from above, which doesn't appear to be the case.
– user9278
Apr 30, 2017 at 8:05