20130906, 15:51  #1 
"Matthew Anderson"
Dec 2010
Oregon, USA
1010000100_{2} Posts 
Triangle puzzle
Hi all,
I found this one interesting. It requires a different way of noodling. Regards, Matt 
20130906, 17:39  #2 
Romulan Interpreter
Jun 2011
Thailand
10001111001001_{2} Posts 
The triangle problem is simple, it is just a trick. The points on the "hypotenuse" of the "triangle" where the small triangles intersect are not collinear, you are not looking to a triangle, but to some "belly", once is concave, once is convex, that is where the additional area come from (which can be easily proved). It never "stumped" me, and I wasn't older then 15 when I saw this problem, more then 30 years ago.
The one with money is more intriguing, and because I was lazy to compute (here is 0:39 in the night!), I did fast a VBA macro in excel, hehe... If you allow M to be zero there are more solutions, otherwise the solution is unique (of course, assuming different letters mean different numbers, otherwise there are thousands of trivial solutions). 7653+0816=08569 7429+0814=08243 7539+0815=08354 8432+0914=09346 8542+0915=09457 8324+0913=09237 6419+0724=07143 7531+0825=08356 7534+0825=08359 7316+0823=08139 5741+0637=06478 5742+0637=06479 6524+0735=07259 6415+0734=07149 3719+0457=04176 3712+0467=04179 9567+1085=10652 Last fiddled with by LaurV on 20130906 at 17:43 Reason: spoilers, if someone wants to play with numbers 
20130906, 17:44  #3 
May 2013
East. Always East.
11×157 Posts 
The two small triangles are not similar triangles, though they are very close. The one is five wide by two high and the other is eight wide by three high.
The top "triangle" is thus slightly concave and the bottom one is slightly convex. The extra "bulge" comes from the extra square. 
20130906, 19:16  #4 
1976 Toyota Corona years forever!
"Wayne"
Nov 2006
Saskatchewan, Canada
13×349 Posts 
My mind comes up with an answer a little different. Convex/concave is not necessary.
If we assume the top triangle is exactly 5X13 it has an area of 32.5 units. If we further assume all vertical and horizontal cuts are exactly on grid lines then simple math shows that the triangle pieces are not exactly 8X3 and 2X5 as they would appear. So when you interchange them.....I think you can take it from here. 
20130907, 00:26  #5 
"Matthew Anderson"
Dec 2010
Oregon, USA
2^{2}·7·23 Posts 
Thanks for all the good answers, guys.
I have simplified the problem to show the angle of the two parts of the hypotinuse. For what its worth. Regards, Matt 
20130907, 02:00  #6 
1976 Toyota Corona years forever!
"Wayne"
Nov 2006
Saskatchewan, Canada
13×349 Posts 
Again I don't think the triangles need to be convex or concave.
If we can trust the picture in the puzzle book and hold a straight edge over the hypotenuse it is straght. Let's carry on: If we assume the top triangle is exactly 13 X 5 it has an area of 32.5 square units. If we further assume all vertical and horizontal cuts are exactly on grid lines then simple math shows that the triangle pieces are not exactly 8X3 and 2X5 as they would appear. They would be 8 X (8*(5/13)) = 8 X 3.08 2 X (2*(13/5)) = 2 X 5.2 And yes their cuts would overlap slightly where they connect near the middle of the hypotenuse. Now if we swap the triangles pieces (and rearrange the other pieces as required) we have a new larger triangle that is (8 + 5.2) X (2 + 3.08) = 13.2 X 5.08 = 33.5 square units requiring the one blank square. And again where the two smaller triangle pieces meet they would not exactly match. Ok so maybe the new triangle would not be exactly straight on the hypotenuse because it is now slightly longer on both the X and Y axis. Last fiddled with by petrw1 on 20130907 at 02:04 
20130907, 03:52  #7 
May 2013
East. Always East.
11010111111_{2} Posts 
Nononono. The smaller triangles are straight. The larger one is concave or convex.

20130907, 03:56  #8 
1976 Toyota Corona years forever!
"Wayne"
Nov 2006
Saskatchewan, Canada
13×349 Posts 

20130907, 04:18  #9 
1976 Toyota Corona years forever!
"Wayne"
Nov 2006
Saskatchewan, Canada
13·349 Posts 
Oh I can see how it can work if the small triangles are drawn slightly off skew so they meet exactly and so that when moved the make the triangle bigger.
HOWEVER.... can it not also work as I suggested? Or am I missing something? It is a few years (ummm decades) since high school. 
20130907, 18:07  #10 
May 2013
East. Always East.
11·157 Posts 
I think your way is good too. It's just more of a numbers approach.

20130907, 21:05  #11 
"Matthew Anderson"
Dec 2010
Oregon, USA
2^{2}×7×23 Posts 
Hi guys and gals,
I made a picture similar to another one shown in this thread. Cheers, Matt 
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