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Old 2013-09-06, 15:51   #1
MattcAnderson
 
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"Matthew Anderson"
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Default Triangle puzzle

Hi all,

I found this one interesting. It requires a different way of noodling.

Regards,
Matt
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Old 2013-09-06, 17:39   #2
LaurV
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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

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Old 2013-09-06, 17:44   #3
TheMawn
 
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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.
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Old 2013-09-06, 19:16   #4
petrw1
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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.
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Old 2013-09-07, 00:26   #5
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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
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Old 2013-09-07, 02:00   #6
petrw1
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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.

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Old 2013-09-07, 03:52   #7
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Nononono. The smaller triangles are straight. The larger one is concave or convex.
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Old 2013-09-07, 03:56   #8
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Quote:
Originally Posted by TheMawn View Post
Nononono. The smaller triangles are straight. The larger one is concave or convex.
Hey I thought us Sask boys stuck together?
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Old 2013-09-07, 04:18   #9
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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.
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Old 2013-09-07, 18:07   #10
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I think your way is good too. It's just more of a numbers approach.
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Old 2013-09-07, 21:05   #11
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Hi guys and gals,

I made a picture similar to another one shown in this thread.

Cheers,
Matt
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