A Fistful of Brain Teasers

Posted by: Don Atkinson on 13 November 2017

A Fistful of Brain Teasers

For those who are either non-British, or under the age of 65………. The UK used to have a brilliant system of currency referred to as “Pounds, Shillings and Pence”. Simplified to £ ״ s ״ d. No! Don’t ask me why the “Pence” symbol is a “d”, just learn it and remember it !

A £ comprised 20 Shillings and a Shilling comprised 12 Pence. Thus a £ comprised 240 Pence. I reckon that both Microsoft and Apple would have difficulty with these numbers in their spreadsheets, more so if we included Guineas, Crowns, Half-Crowns and Florins. However, I digress..............

The purpose of the explanation is to assist with the first two or three teasers that follow. So just to ensure a reasonable comprehension has been grasped…. ….. if each of three children has £3 − 7s − 9d, then collectively they have £10 − 3s − 3d Got the idea ? Good ! Just try 5 children, two each with £4 − 15s − 8d and three each with £3 − 3s − 4d. How much do they have between them ? (this isn’t the first brain teaser, just the basic introduction with some “homework”, the Teasers follow)

Posted on: 01 May 2018 by Don Atkinson

Hi ITBUFFSTER,

Yes, this is a problem that is a bit more difficult to solve than might first appear.

Your solution is interesting and not one that I have seen before. Nonetheless, it should work a treat.

I note that you have used “d” to differentiate the length of ladder above the chest, and the length below the chest. The original question uses “d” to define the required solution for the distance of the foot of the ladder from the chest – you have used “x” for this dimension. No problem, I just mention it for clarity, especially for others who might be following your solution. In a similar vein, you have used “2L” for the length of the ladder as opposed to “L” as per the teaser text. Again, no problem, but we need to be aware.

I tested my own solutions (I have three solutions) by drawing graphs in Excel. They all co-incide. And the solutions to specific lengths of ladder, eg 8m, all co-incide exactly. In this instance d (or in your terms “x”) = 0.144557m. For a 5m ladder, my “d” would be 0.2605m (your “x”)

I have had a little difficulty with your formula for “d” (or d² ). When I substitute L=4 (Ladder length 2L = 8m) I get unrealistic answers. Could you check your formula for d² and (in particular) clarify the power of the element “4L”. I have presumed the power is “4” when a=1 (as per the teaser).

I have no doubt your solution works, but tonight I do seem to be suffering from some form of algebra-blindness !

Cheers, Don

Posted on: 01 May 2018 by itbuffster

Hi Don

Yes, no harm in clarifying notation. I used a ladder length of 2L to save writing L/2 all the time.

The formula for d^2 works if you use a minus sign between the two terms (I originally wrote + or - because it was the solution of a quadratic equation). It then reads:

d^2 = (L^2 + a^2) - sqrt(a^4 + 4*a^2*L^2)

In the square root sign, the first term is a raised to the power of 4, and the second term is 4 x L squared x a squared.

For any length of ladder there are two solutions (as long as the ladder isn't too short). These correspond to the ladder making an angle of A degrees or (90 - A) degrees with the ground.

My solution generates x values (distance of the foot of the ladder from the edge of the chest) that *decrease* as the ladder gets longer - in other words, the angle the ladder makes with the ground increases.

I agree that when the ladder is 8m long, x (your d) is 0.14457m but I get x = 0.111882m when the ladder is 10m long (less than the former case, as expected).

I'll be interested to see your solutions when you're ready to share them. It would be nice if there was a really elegant way to crack this.

Posted on: 02 May 2018 by Don Atkinson

itbuffster posted:
Hi Don
Yes, no harm in clarifying notation. I used a ladder length of 2L to save writing L/2 all the time.
The formula for d^2 works if you use a minus sign between the two terms (I originally wrote + or - because it was the solution of a quadratic equation). It then reads:
d^2 = (L^2 + a^2) - sqrt(a^4 + 4*a^2*L^2)
In the square root sign, the first term is a raised to the power of 4, and the second term is 4 x L squared x a squared.
For any length of ladder there are two solutions (as long as the ladder isn't too short). These correspond to the ladder making an angle of A degrees or (90 - A) degrees with the ground.
My solution generates x values (distance of the foot of the ladder from the edge of the chest) that *decrease* as the ladder gets longer - in other words, the angle the ladder makes with the ground increases.
I agree that when the ladder is 8m long, x (your d) is 0.14457m but I get x = 0.111882m when the ladder is 10m long (less than the former case, as expected).
I'll be interested to see your solutions when you're ready to share them. It would be nice if there was a really elegant way to crack this.

Ah ! in the square root sign, the second term is 4 * L² * a²........................got it !

So for L =4 (ie an 8m ladder) and a = 1 (the side of the chest), this second term is = 64 and (your) "d" = 2.989606

I had previously read your solution as sqrt [a⁴ + ((4L)^2a)²]

ie sqrt [1 + 16⁴] when a=1

My distance "d" 0.2605m was for a ladder 5m long

For a ladder 10m long, the distance is 0.1119m ie the same as your 10m ladder.

Yes, there are always two solutions, one for the "upright" ladder and one for the "flat" ladder. In fact, the two solutions simply provide the horizontal distance from the chest to the toe of the ladder and the vertical distance from the top of the chest to the head of the ladder.

I'll post my solution later, but remember, the original question set the chest (cube) as side "unity", which makes the arithmetic and solution a bit more simple.

Posted on: 02 May 2018 by itbuffster

Cheers Don

I agree with all that. Here are the results from my Excel spreadsheet:

2	0.3622
2.5	0.260518
3	0.205013
4	0.144557
5	0.111882
6	0.091322
7	0.077169
8	0.066825
9	0.058932
10	0.052708

The first column is my L (so double it to get the length of the ladder). The second column is my x (your d).

Looks like I get the same answers as you do.

Posted on: 02 May 2018 by Don Atkinson

itbuffster posted:
Cheers Don
I agree with all that. Here are the results from my Excel spreadsheet:
2 0.3622
2.5 0.260518
3 0.205013
4 0.144557
5 0.111882
6 0.091322
7 0.077169
8 0.066825
9 0.058932
10 0.052708

The first column is my L (so double it to get the length of the ladder). The second column is my x (your d).
Looks like I get the same answers as you do.

Well done.

I do like your general formula which incorporates "a", ie the dimension of the cubic chest.

The minimum length of ladder that fits the bill is 2√2 or in your case, such that "L" ie the half length is √2

In this case the toe of the ladder is 1m away from the base of the chest, as you noted previously.

My answers are the same as yours viz (L; d)

3; 0.6702 4; 0.3622 5; 0.2605 6; 0.2050 7; 0.1694 8; 0.1446 9; 0.1261 10; 0.1119

15; 0.0716 20; 0.0527 etc

Posted on: 02 May 2018 by Don Atkinson

The most straightforward formula that I have, (like yours best expressed in two parts) is as follows:-

d = ½[y ± √(y² - 4)]

y = [√(L² + 1)] - 1

Note "d" is the distance required ie from the base of the chest to the toe of the ladder. You need to take the "minus" option if the ladder is "upright"

L is the length of the ladder (not the half length)

Posted on: 02 May 2018 by Don Atkinson

Perhaps a more "concise" solution is :-

d = e ^{–Acosh(y/2)}

where "y" has the same meaning as in my earlier post.

Posted on: 02 May 2018 by itbuffster

That looks really neat Don, but what is A?

Posted on: 02 May 2018 by Don Atkinson

A as in Acosh is the inverse hyperbolic cosine.

Similar to Atan, Asin, Acos etc ie inverse tangent, inverse sine, inverse cosine.

I think Excel uses Acosh, Atan etc. Some programs (and possibly Excel) use arc tan etc

hope this helps.

Posted on: 16 May 2018 by Don Atkinson

This is the first of my three solutions to the "ladder" problem, in graphical format.

Posted on: 16 May 2018 by Don Atkinson

This is the second solution. You will note that the plot is identical to the first solution, despite the different input.

You will notice that the plot values are based on integer values of ladder lengths "L" as they were in solution 1 above ("L" is plotted on the X axis; "d" is plotted on the Y axis)

Posted on: 16 May 2018 by Don Atkinson

And this is the third solution.

OK, it's not quite a real solution, but it's easy enough to input successive values of "d" until you get the "given" ladder length.

So, although the graph is clearly the same curve as solutions 1 and 2, the actual plot values are based on given values of "d" (on the Y Axis)

Posted on: 03 July 2018 by Don Atkinson

There are many puzzles and situations in life with complex solutions, which can only be arrived at in stages. A methodical approach, designed to narrow the search in successive steps as per detective work or a forensic exercise is often successful. For example :-

An oil executive lives on the twenty fourth floor of a tall apartment building in Dubai. Each morning when he goes to work he calls the lift, pushes the ground floor button and is driven by chauffeured limo to his downtown office.

On his return home, he enters the lift, usually pushes the twelfth floor button and walks up the rest of the stairs. At other times he rides straight up to his floor.

Why does he not always take the lift to the twenty fourth floor ?

The ground rules for these puzzles are simple, the solution must :-

fit all the given facts

conform to acceptable norms of behaviour

obey the laws of the physical world as we know them

I'll outline a typical solution later, once you've all had time to consider, an if necessary clarify the puzzle concept.

Posted on: 03 July 2018 by Minh Nguyen

The lift button panel could look like this:

24 25

22 23

20 21

...

12 13

10 11

8 9

6 7

4 5

2 3

G 1

The person could be very small, so on the way up, the highest button they can reach is the button for the twelfth floor. On the way down, they can always reach the button for the ground floor.

Posted on: 03 July 2018 by Eoink

I can't help wondering if the 13th floor is relevant here.

Posted on: 03 July 2018 by Beachcomber

Minh Nguyen posted:
The lift button panel could look like this:
24 25
22 23
20 21
...
12 13
10 11
8 9
6 7
4 5
2 3
G 1

The person could be very small, so on the way up, the highest button they can reach is the button for the twelfth floor. On the way down, they can always reach the button for the ground floor.

Why not press the 13? No more difficult than the 12.

Possibly worse still for this solution depends on whether in Dubai they have a G, or whether as in America the first floor is what in the UK (and Europe, IIRC) is the Ground floor, in which case he might only be able to reach the 12. Or the joke started out in the US, with no G button.

Posted on: 03 July 2018 by TOBYJUG

The oil executive ALSO lives in a private penthouse of a 12 storey tall apartment.

Posted on: 03 July 2018 by Minh Nguyen

Beachcomber posted:
Minh Nguyen posted:
The lift button panel could look like this:
24 25
22 23
20 21
...
12 13
10 11
8 9
6 7
4 5
2 3
G 1

The person could be very small, so on the way up, the highest button they can reach is the button for the twelfth floor. On the way down, they can always reach the button for the ground floor.
Why not press the 13? No more difficult than the 12.
Possibly worse still for this solution depends on whether in Dubai they have a G, or whether as in America the first floor is what in the UK (and Europe, IIRC) is the Ground floor, in which case he might only be able to reach the 12. Or the joke started out in the US, with no G button.

Perhaps I should have defined the lift button panel as follows:

23 24

...

11 12

...

G -1

Posted on: 03 July 2018 by Mulberry

The second approach from Minh seems to be right when the executive pushes the button himself. At the other times another taller person is with him in the elevator and pushes the top floor button for him.

“On his return home, he enters the lift, usually pushes the twelfth floor button and walks up the rest of the stairs. At other times he rides straight up to his floor.”