A Fistful of Brain Teasers

B)

Because i) The net displacement hasn’t changed: the boat + dead hubby and the 135s displace the same amount of water whether the body and 135s are in the boat, dangling below it, or released into the depths. 

or ii), even if that were not the case the depth change would unmeasurable small given the vast volume of the reservoir compared to that of the dumped body, unless it the reservoir really very tiny, barely bigger than the boat!

I trust the 135s were well sealed in plastic ... and that you have a record of the precise location where this happened.

Water level falls. When the macabre denser-than-water object is in the boat, it displaces a volume of water that equals its weight. Once submerged, it displaces only the smaller water volume of its own volume.

Now I’m kicking myself - but of course she would not see any change in the water level indicator, as indicated by ii) in my response...

Mulberry posted:

Hi Don,

the water level goes down, which is not what I thought at first. Both my wife and I liked this one, so I had to try it out in our kitchen (wife and Hi-Fi are completely unharmed). Not sure why this is the case at the moment, although I have an idea which I want to think over.

Hi Mulberry,

I'm pleased that both you and Mrs Mulberry liked this one.

There is nothing like an experiment to get the right answer is there

In fact, most if not all, of our knowledge comes from experiments. We simply devise formulae to help us repeat and and predict the outcome of further experiments and activity.

Well done !

winky also has the right answer and a clear, concise explanation.

Innocent Bystander posted:

B)

Because i) The net displacement hasn’t changed: the boat + dead hubby and the 135s displace the same amount of water whether the body and 135s are in the boat, dangling below it, or released into the depths. 

or ii), even if that were not the case the depth change would unmeasurable small given the vast volume of the reservoir compared to that of the dumped body, unless it the reservoir really very tiny, barely bigger than the boat!

 

I trust the 135s were well sealed in plastic ... and that you have a record of the precise location where this happened.

Hi IB,

Nice to see you got there in the end !

And as you pointed out, in all realistic scenarios of reservoir/boat/body you wouldn't detect any movement against a water mark.

winkyincanada posted:

Water level falls. When the macabre denser-than-water object is in the boat, it displaces a volume of water that equals its weight. Once submerged, it displaces only the smaller water volume of its own volume.

Nice one winky,

and a nice concise and clear explanation.

Hopefully, this diagram might help anybody still struggling.....

It should be easy enough to follow Left --> Right ie Part 1 --> Part 2 --> Part 3 (assuming you are familiar with the principle involved in Part 3.)

The "displacements" in Parts 2 and 3 are relative to Part 1

The Brain Teaser simply "starts" at Part 3 and asks you to work back to Part 2

Farmer Watt’s Estate

Down in rural Wiltshire, my friend Farmer Watts owned three square fields as shown in the map. The field were 26, 20 and 18 acres in size.

He bought the four intervening triangular fields last year in order to consolidate his holding.

What is the combined acreage of his seven fields ?

It’s possible to solve Farmer Watts “Teaser” in more ways than one ! I can hint at two !

First is the “hand-grenade + Excel” technique !

Second is far more subtle, but only works easily because the numbers have been carefully selected – as per many Brain Teasers !

For the first you will need to recall :-

• Heron’s Formula regarding triangles in which all three sides are known
• Law of cosines, again in triangles in which all three sides are known
• The SAS formula for triangles in which you now know two sides and the included angle

It will solve any collection of fields similar to those in this teaser, but of different areas. Clearly it's no different to a bit of A-Level maths.

And in Excel it works out quite easily, and rather precisely !

For the second method , you just need a tiny bit of Pythag and a keen eye……but then, that's what "Teasers" are all about

This is just a 60 seconds "filler" but be honest with yourself......

....was yourfirst instinctive  answer correct ?

Persevering with the "hand-grenade" technique....

Basically, in any triangle ABC with angles A, B and C and sides a, b and c

Heron’s formula……Area = [s(s-a)(s-b)(s-c)]^0.5          s = semi-perimeter

Law of Cosines……..    a² = b² + c² – 2bcCosA          b² is similar as is c²

SAS formula………...Area = (abSinC)/2                     or  (acSinB)/2     or (bcSinA)/2

You now have everything you need to calculate the areas of the four new fields but....

....you do need a calculator or tables or a computer and a bit of careful calculation.....

and i'm sure you didn't really need me to remind you about these formulae

or....you could look at the three initial square fields more carefully because those areas are not just random numbers and I drew the 26 acre field "skew-whiff" on purpose

The second technique is more subtle than the first, but much easier to carry out without a computer....

I have added a "Grid" to the initial drawing.

I have accurately drawn the sides of Field A as √26.

I did this by using the Hypotenuse of a triangle where the other two sides are 5 units and 1 unit

I doubt if I can provide much more of a "hint" without solving the problem directly

Don Atkinson posted:

This is just a 60 seconds "filler" but be honest with yourself......

....was yourfirst instinctive  answer correct ?

So how many of us initially (ie, almost instinctively and immediately) thought "10P" ?

OK, I know nobody is going to admit to this, but ask a few friends................

Oh well, I sure thought that immediately after reading the teaser. Now half the amount seems a much better answer 

Mulberry posted:

Oh well, I sure thought that immediately after reading the teaser. Now half the amount seems a much better answer 

You're not alone Mulberry !!

The red rectangle should help visualize how I created the three square fields.

5² + 1² = 26²

4² = 2² = 20²

3² + 3² = 18²

The area of the red rectangle is 5 x 4 = 20 acres.

This comprises Field D plus three red-sided triangles

Deduct the three red-sided triangles outside area D and you have the area of Field D

It shouldn't be too difficult to sort out the areas of the other three fields that farmer Watts bought.

Take the sticks off one at a time, such that you are always removing the top one off the pile. What order do you remove the sticks ?

PS There are no "tricks" !!

I would approach it this way: G - A - F - D - E -  B - H - C

The right answer Mulberry, for removing the sticks.

I presume you (and a few others) have by now, got the area of the triangular field D - and possibly the other three triangular fields ?

Area of D is 10 acres i.e. the area of the 5 x 4 rectangle less the combined areas of the 3 triangles calculated by  using their height x half the base. {2.5, 3.5 and 4)

Hi sjb,

you are certainly on the right track. And almost there.

if you look slightly more closely at the red rectangle and the three triangles with red sides you will see that these red sides are 5*1; 3*3 and 4*2. 

the areas of theses triangles doesn’t add up to 10. Close ! But not quite close enough. So triangle D isn’t quite 10 acres. 

I bet you can soon rectify the arithmetic.

The other 3 fields can each be seen to be within similar rectangles with surplus area to be deducted.

so, almost there ! 

Nope. I think I’ll just accept that I’m thick.

Triangle D is 9 acres. 20 - (2.5 + 4.5 + 4).

sjbabbey posted:

Nope. I think I’ll just accept that I’m thick.

Far from thick sjb !

you had the right concept ie [Rectangle - (3xTriangles)]

Even the right basic maths ie [area = ½*base*height]

It was just a matter of ½*5*1 + ½*3*3 + ½* 4*2.

Giving 2½ + 4½ + 4 (rather than 2½ + 3½ + 4)

But the right concept was the breakthrough !

Hungryhalibut posted:

Triangle D is 9 acres. 20 - (2.5 + 4.5 + 4).

 

Spot-on HH

Only three more fields to go....................

....should be a doddle for a "retired" accountant !! Just think of it as West Sussex CC buying up land for a few cycle parks

and winky would be in there if the units were sq miles and the purpose was open-cast coal mining !