Brain Teasers are Back!

It’s beginning to look like I had better type up my workings that lead to the solution… :sunglasses:

*PS. For anybody why might have given it a try, the answer is 8 yards per second. *
Now all you need do is link 4 yards, 3/4, 12 spokes, 24 spokes, 16 frames/sec, and the answer 8 yards/sec Enjoy, even if you don’t wish to post !

I have typed up my working for the solution to the Stagecoach Wheels …

I hope it makes sense !

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Grid 1 to 9 General Instructions.

There are six equations in the grid below.

Using the numbers 1 to 9, once each, complete the equations.

I have given you a start by inserting some of the numbers.

The arithmetic needs to be done in the order in which the numbers appear eg 1 + 2 x 3 would equal 9 rather than the mathematically correct way whereby 1 + 2 x 3 = 7.



A Shade Bigger

This puzzle is from Mrs R.

What proportion of the large square is shaded?

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Well, it looks like it’s half … but …
I think the answer is in the title of your post !

That square on the slope is a shade smaller than a quarter.
I’ll leave the precise “shade” for others, for the moment :sunglasses:


Not quite right but you are on the right lines

Miscounted…9/17ths. Hmmm, cannot be an odd denominator?

You are right that the denominator is even and also that it works out to be a reasonably simple fraction.

My number was 19/36.

Which is “a shade bigger” than a half.

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Spot-on Steve
Well done.

The figure was derived from 1 - (1/4) - (2/9) = 19/36
“1” being the enclosing large square
1/4 being the proportion of the bottom-left white square
2/9 being the proportion of the (smaller) white square sitting on the main diagonal.

The smaller white square has side length = 1/3 the diagonal of the enclosing large square. The two congruent, right-angled, isosceles triangles, either side of this smaller white square, provide confirmation of the size of the smaller white square.

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Don, that gives the right answer. There are other methods too:

Firstly Pythagoras. As you noted, the white square has side length 1/3rd of the diagonal of the main square (because the two shaded triangles either side of the white square have the same side length). Hence this white square has side length (√ 2)/3 if big square is 1x1. Area is 2/9 and the other white square is 1/4. I think this is similar to your solution, but maybe expressed differently.

Secondly Origami. Fold the shaded triangle in the top right “envelope style” over the white square. It covers one quarter of it. The other two triangles fold over to each cover half the square. Hence if the area of the top right triangle is 1, the shaded area is 1 + 2x2 and unshaded is 4, so the shaded proportion of top right is 5/9 (=10/18). In the bottom left, the shaded triangles can fold over to completely fill the white square, so the proportion is 1/2 = 9/18. Overall this gives us 19/36.

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9 blue, 8 white
Clearly I’m no mathematician :smile:

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Thought it might help if I posted the sheet on which I did my working out !

You might need to click on the attachment to enlarge it and make it readable.

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Ah, I see.

The triangles in the top right are smaller than the bottom left (there are 9 vs 8 in each half of the square).

Hi badyogi, the triangles in the top right are not the same size as those in the bottom left.

The 8 in the in the bottom left have sides 1/2; 1/2 and (Rt2)/4
The 9 in the top right have sides 1/3; 1/3 and (Rt2)/3

However, they are all Right-angled Isosceles Triangles, so the geometry is fairly straightforward.

Total area Top Right (there are 9 triangles) = 9 x (1/2 x 1/3 x 1/3) = 1/2
Total area Bottom left (there are 8 triangles) = 8 x (1/2 x (Rt2)/4 x (Rt2)/4) = 1/2

The area of each triangle in the top right = (1/2)/9 = 1/18
The area of each triangle in the bottom left = (1/2)/8 = 1/16

Thea area (shaded proportion) that is required = (5 x 1/18) + (4 x 1/16) = (10 + 9)/36 = 19/36

Your diagram (and Ravvie’s origami approach) leads to the solution by a far more elegant route than my initial ‘heads-down’ roaring-bull type of approach :sunglasses: Well done !

As 9/17 = 0.529 and 19/36 = 0.528, I think you get the prize for an answer that is “a shade bigger” than the mathematically correct one!