We had two very different approaches in our household to the ladder in a well teaser. We both arrived at the same solution though.
I will wait for you to post your workings.
I think the result in this situation would result in more tea in the milk, than there was milk in the tea.
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I agree with both of your supplementary answers.
I had to think about it as I was sipping my tea.
I started with equal volumes of tea and milk.
My ‘teaspoon’ put 20% of my milk into my tea.
My ‘sip’, swallowed one third of my now milky tea.
I then returned the ‘teaspoonful’ of now milky tea, to the milk cup.
I suppose I should replace specific volumes, or proportions, with generalised proportions, but ….
Steve’s answer of 10m is correct. As is Ravvie’s.
Well done.
One of us should post at least one solution …
A tall tree in a remote forest falls to the ground. There is nobody around. The question is ‘does it make any sound?’
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OK, my approach is as follows. I suspect Ravvie or Mrs R has a neater solution.
(Apologies that I don’t know how to show numbers to a power properly on here 
).
The wells are cylinders with area = pi x radius squared x H (where H is well depth and also ladder length).
The first well radius must therefore be 0.5m, ie square root of 0.25. With a height of 5m, its ratio of height to radius is therefore 10:1.
I took Mrs D’s instruction for Well 2 to be “mathematically similar”, to mean that this 10:1 ratio should be maintained.
So, pi x radius squared x H = 10 x pi, but also H must equal 10 x radius.
So, pi x 10 x radius cubed = 10 x pi.
So radius cubed = 1, and therefore radius = 1m.
Height = 10 x radius, which is 10m.
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Nice answer! Exactly as Mrs R did it ( in her head of course, as always).
Mrs R asked me for my answer. I’m a bit more maverick at solving problems (Mrs R prefers rigour throughout).
I said it is eight times the volume (10 vs 1.25), so that’s double in each of three dimensions. Giving 10 metre length.
I then went on to say that the shape is irrelevant, it could be square, starfish shape or even shaped like the Statue of Liberty. Big mistake being so cocky, I had to find a large hole in the garden to hide in. Anyone got a 10 metre ladder?
You are very good at using “intuition” (for want of a better word) to find neat shortcuts.
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Scale factors. Applicable to ‘mathematically similar’ one-dimensional; two-dimensional and three-dimensional shapes.
If a one-dimensional feature (eg a straight line) is doubled in size, the new length is 2 x the old length
If a two-dimensional area (eg a square) is doubled in length, the new area is (2^2) x (the old area). ie 4 x old area
If a three-dimensional volume eg a cube, or a cylinder) is doubled in length, the new volume is (2^3) x (the old volume). ie 8 x the old volume.
10pi divided by 1.25pi = 8
Cube root of 8 = 2
depth, width, breadth of my well will each be twice as large as my neighbour’s well.
The well isn’t necessarily circular in cross section. But the use of ‘pi’ in the teaser leads some of us (that included me !) in that direction !
Now, if we all keep quiet, very quiet, we might hear that tree falling down …
Well done Steve. Well done Ravvie. And well done Mrs R.
I’ll give it a shot …
To my mind, the answer to this question depends on the definition of sound.
When a tree falls, the motion disturbs the air and causes air vibrations. This physical phenomenon, which can be measured by instruments (other than our ears), exists regardless of our perception (hearing or seeing) of it.
We can (if we so wish) define sound as our perception of such air vibrations. Therefore, sound does not exist if we do not hear it.
In other words, although the falling tree sends off air vibrations, it does not produce sound if no human is within the distance where the air vibrations are strong enough for a human to perceive them.
However, if we define sound as the air vibrations themselves, then sound would be produced.
I bet the Internet is full of interesting debate …
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Good shot Don. Makes perfect sense to me!
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A very good explanation Don.
Having read John Atkinsons (former editor of stereophile and still in charge of measurements) Heyser lecture on the magazine’s website a few months ago helped me to better understand that (and some other things).
The frequencies defined as sound for humans, or at least some of them, are outside of the hearing limits of other species. Just as infra- and/or ultrasound is within other species hearing range.
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Rectangle Diagonal ?
Some of you might have noticed that I occasionally present my Brain Teasers, or my answers, using squared paper.
I have outlined a 12 x 8 Rectangle below, using just such squared paper.
If I draw a diagonal of this rectangle, how many squares will it cross ?
You travel through 4 3 x 2 rectangles which are diagonally adjacent. In each, 4 squares are crossed, for a total of 16 squares crossed in the 12 x 8 rectangle.
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Well done JR. And a wonderfully clear explanation too.
I’ve also added an update to my initial diagram.
When you said squares I thought you meant any size of square. The diagonal crosses several 2x2, 3x3 etc squares.
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Ah!
On re-reading the initial teaser, I realise I could have (actually should have) been far more careful with the description of “squares”
Apologies.
But I think you have just created an even better (more difficult) teaser
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Bugger.
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