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

*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 !

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.

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.

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.

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 Well done !