Brain Teaser No 1

Don,
Your question about leading digits is refering to Benford's Law which has been used to spot accounting fraud. The answer is d).
John

quote:

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Good answer Two-Sheds

Does anybody want to post an explanation/set of formulea....

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Is this like being back at school, I have to my working to get any marks

M = mans age, W = wifes age, Q = years married (all at present time).

man is 6 years older than wife:
(1) M = W+6
4 years ago man had been married half his life:
(2) Q-4 = 1/2(M-4)
in 10 years wife will have been married two thirds of her life:
(3) Q + 10 = 2/3(W+10)

takin the value for M in terms of W from (1) and placing into (2) we get:
Q-4 = 1/2(W+6-4)
Q = 1/2(W+2) + 4

take this value of Q and substitute into (3):
1/2(W+2) + 4 + 10 = 2/3(W+10)

multiply both sides by 6 to get rid of fractions:
3(W+2) + 84 = 4(W+10)
3W + 6 + 84 = 4W + 40
3W + 90 = 4W + 40
W = 50

substitute back into (1):
M = 50 + 6 = 56
substitute back into (2):
Q-4 = 1/2(56-4)
Q-4 = 26
Q = 30

Don phrased the question thus,

quote:

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Assuming that such tables comprise RANDOM numbers, (yes, I know that in real life they aren't random, but let's assume they are) what is the probability that a number in such a table has a leading digit of 1 eg 10,600; 1,250,000

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So 'Benford's Law' cannot apply and the answer is 11.1%

Paul

Steved, John, Paul,

oh dear, looks like I dropped a clanger again.

I introduced the word RANDOM into the "list" teaser, to make it 'clear' that the lists were genuine and not 'doctored'

I had in mind Benford's Law, although I didn't know it was called Benford's Law!

However, it seems that in writing the word 'random', I have changed the whole concept.

Thanks to you all for having a go and managing to find my mistakes, as well as the right answers.

Cheers

Don

Two-Sheds

Is this like being back at school, I have to my working to get any marks

9/10

Cheers

Don

Omer,

This Benford law is quite amazing.

I am trying to see if it stems simply from exponential distribution which shifts the probability towards lower numbers.

My recollection is that the probability of a number starting with 1; 2; 3; etc decreases towards 9

I somehow remembered that the probability of 1 was 30.1%.

Whether the rest follow an exponential curve or some other smooth decrease I am not sure.

Perhaps John or Paul or..... could help out

Cheers

Don

This site has an explanation of Benford's Law as well as a lot of other good mathematical stuff.
John

One for the rainy days...

If the probability of rain on any given day on a certain factory in Salisbury is 50 percent, what is the probability that it will rain on the factory for exactly 3 days in a five day period.

Cheers

Don

another rainy day....

If the probability of rain on any given day on a certain factory in Salisbury is 50 percent, what is the probability that it will rain on the factory for at least 3 days in a five day period.

Cheers

Don

actively trolling....

no need to actually wait for a rainy day.....before having a crack.

even if they are all too easy these days

Cheers

Don

There are 2^5 = 32 possible periods. The chances of any particular combination is equal.

1 is all wet.
5 have one dry day.
10 have 2 dry days, and 3 wet.

so the chances of rain for exactly three days is 10/32 or 5/16. It follows that the chances of rain for at least three days is (1+5+10)/32 or 1/2.

Paul

Nicely done Paul,

even if they are all too easy these days

Cheers

Don

People who live in glass houses.....

Alex gets into a glass lift on the 11th floor and rides up at the rate of 57 floors a minute. At the same time Matthew gets in the adjacent glass lift on the 51st floor and rides down at the rate of 63 floors a minute. At which floor do they pass each other.

Cheers

Don

30th?

Dan,

lets see if anybody agrees.....

cheers

Don

lets see if anybody agrees.....

I always get a sinking feeling when you say that

Naim Packaging Manager required

7 days a week, 364 days a year (Xmas holiday)

Minimum wage honoured.

We are looking for a new packaging manager for our new pre-assembled Main Fraim department. No more flat-pack DIY stuff, so our ex MFI manager had to go......


All candidates must be able to answer the following entry exam question.....

The new Main Fraim packaging will be made from a thin, square sheet of aluminium with sides each 5m long. The open-top box will be formed, in effect, by cutting a square from each corner, folding up the sides and jointing the cut edges. What is the maximum capacity of the box that can be made ?

Cheers

Don

I always get a sinking feeling when you say that

you shouldn't, you're in the lift going up....

cheers

Don

Main Fraim

I say 4.63 cu meters approx., or 5/3 of a meter on a side.

cheers

Dan

For the fraim box, I think the dimensions of the box would be 5/6 by 10/3 by 10/3 giving a volume of 250/27 square metres or 9.26 square metres.

proof (to keep don happy):

If we call the length of the side to be cut x then the volume of the box can be given by:
(5-2x)(5-2x)x = 4x^3 - 20x^2 + 25x

If you draw a graph of x against volume then you will get a curve and when the box is at it's minimum capacity and maximum capacity the curve will be flat (i.e. have a gradient of 0). To get the formula for the gradient we differentiate the equation so we get:

dy/dx = 12x^2 - 40x + 25

as mentioned above we want to find x when dy/dx is 0:

0 = 12x^2 - 40x + 25

we can solve this using the standard equation for solving quadratic equations:

ax^2 + bx + c = 0

x = (-b +- sqrt(b^2 - 4ac)) / 2a

x = (40 +- 20) 24

so x can be 5/2 or 5/6. If x is 5/2 or 2.5 the dimensions end up being 2.5 by 0 by 0 so the volume is 0 and this is the minimum volume, so for max volume x is 5/6.

proof (to keep don happy):

very good.

Why not 'perfect' ?

slight typo....the VOLUME will be in CUBIC metres rather than square metres...

But thats just nit-picking

Cheers

Don

Dan M,

I trust you accept Two-shed's account of the volume ?

Your 30th floor is of course right. It was only because you followed it with a ? that pushed me to tease....well it is a brain 'teaser' thread.

For those who just browse here, the lifts start 40 floors appart [51 - 11] and have a closing speed of 120 floors per minute [120 = 57 + 63] (ok, speed is usually mph or kph etc but there's nothing wrong with floors per minute....)

Hence they will travel the combined 40 floors and meet after one third of a minute (or after 20 secs as Omer helpfully wrote).

After 20 secs (1/3 minute), Alex's lift will have shot up 57/3 = 19 floors, whilst Matthew's lift will have plumetted 63/3 = 21 floors.

11 + 19 = 30th floor; whilst 51 - 21 = 30th floor also....

Which is where they will pass.

Cheers

Don

Test to Destruction....)

The R&D dept at a dubious hifi manufacturing outfit in Salisbury UK, who shall remain naimless, has devised a simple destructive test for their locally based competitor's preamps.

A mass of 30 kg (the competitor's latest preamp) is placed on a smooth horizontal table. A light inextensible string connects it over a smooth peg to a 50 kg mass (bag of cement), which hangs vertically. Find the tension in the string if the system is released from rest. (Take g = 9.81 m/sec2)

Cheers

Don

Yet another hifi factory in Salisbury....

Matthew R and Mick P have both got Saturday jobs at a hifi factory in Salisbury. Matthew, so he can earn a bob or two to fund his anti-war marching campaigns and Mick so he can buy another gun. Both cycle to/from work....but that's another story....

Paul D has completed his time and motion studies on the pair and found that Matthew takes 8 hours to test 50 Nac 552 preamps whilst Mick can test the same number in 6 hours. How many hours (to 4 sig fig) will it take them to test a batch of 100 between them, starting together, but working independantly. (tea breaks, coffee breaks etc and time out for cosy discussions about politics can be ignored)

Cheers

Don

[This message was edited by Don Atkinson on Sat 29 May 2004 at 20:51.]

quote:

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Originally posted by Don Atkinson:

Matthew R and Mick P have both got Saturday jobs at a hifi factory in Salisbury. Matthew, so he can earn a bob or two to fund his anti-war marching campaigns and Mick so he can buy another gun.

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You don't think Mick is going to shoot Matthew do you? They've been a touch at loggerheads in the past, you know.

JonR