A Fistful of Brain Teasers

Again, a bit of care needed. Some people count it TWICE ! Once counting squares from top-left, diagonally down to the right. Then counting squares from bottom right to top left, they count it again !

A group of people on their Xmas coach outing went into a cafe for a snack. The party leader ordered a cup of tea and a BLT sandwich for everyone (ie one each, not one cup of tea and one sandwich to share - and the number includes the coach driver and the party leader !).

The total bill came to £18.49. (Last of the big spenders !)

How many people were on the coach ?

A fair coin is tossed three times in succession. What is the probability of getting at least one head ?

A fair coin is tossed four times in succession. What is the probability of getting exactly two heads and two tails ?

Hoop La

A thin hoop, diameter d is randomly thrown onto an infinitely large chess board, the squares of which each have side L.

What is the probability that the hoop will lay across more than one square ?

(Assume that d is less than L, ............. if d is larger than L, the probability is 1 ie a dead cert !)

Here's one I can do! The odds of getting at least one head is one minus the chance of getting all tails. The odds of that are 1 in 8, so the answer to the question is 7/8.

JRHardee posted:

Here's one I can do! The odds of getting at least one head is one minus the chance of getting all tails. The odds of that are 1 in 8, so the answer to the question is 7/8.

Whoo Hoo !

Well done and glad you feel included again !

Don Atkinson posted:

A group of people on their Xmas coach outing went into a cafe for a snack. The party leader ordered a cup of tea and a BLT sandwich for everyone (ie one each, not one cup of tea and one sandwich to share - and the number includes the coach driver and the party leader !).

The total bill came to £18.49. (Last of the big spenders !)

How many people were on the coach ?

Ah, I was assuming that I’d end up with 2 answers, but as 1849 is a perfect square, the answer is 43 people, and I’d appreciate the address of the cafe where a cuppa and a BLT cost 43p. (I discarded 1,849 people on the coach at a penny each on the grounds of health and safety.)

Well done Eoink.

Yes, a list of prime numbers came in useful when I first tried it.

You must remember that these puzzles come out of some VERY old O-Level maths books.

The 43p, which was more like 8s''9p would certainly have gotten you a very nice cup of tea and sandwich back then. I could give you the name of the cafe, but I have this funny feeling that £4.30 was what I saw on the menu when I walked past last week ....

Don Atkinson posted:

A fair coin is tossed four times in succession. What is the probability of getting exactly two heads and two tails ?

I'll bring this one and the next one forward before they get lost.

Don Atkinson posted:

Hoop La

A thin hoop, diameter d is randomly thrown onto an infinitely large chess board, the squares of which each have side L.

What is the probability that the hoop will lay across more than one square ?

(Assume that d is less than L, ............. if d is larger than L, the probability is 1 ie a dead cert !)

This is the other one.

Don Atkinson posted:

Don Atkinson posted:

Hoop La

A thin hoop, diameter d is randomly thrown onto an infinitely large chess board, the squares of which each have side L.

What is the probability that the hoop will lay across more than one square ?

(Assume that d is less than L, ............. if d is larger than L, the probability is 1 ie a dead cert !)

This is the other one.

(d/L)^2 - 2d/L 

Merry Xmas - off to start a day of eating now! 

Ian G. posted:

Don Atkinson posted:

Don Atkinson posted:

Hoop La

A thin hoop, diameter d is randomly thrown onto an infinitely large chess board, the squares of which each have side L.

What is the probability that the hoop will lay across more than one square ?

(Assume that d is less than L, ............. if d is larger than L, the probability is 1 ie a dead cert !)

This is the other one.

(d/L)^2 - 2d/L 

Merry Xmas - off to start a day of eating now! 

In my haste to be off for Xmas dinner I think I got the overall sign in the above expression wrong - should be  2d/L - (d/L)^2 . 

Certainly looks better now, Ian !

These are my solutions. Either of which can easily be transformed into Ian's solution.

Different people seem to prefer different presentations.

The second version above shows the probability as 1 - P(Ā) ie One minus the Prob of the hoop NOT laying across more than one square where P(Ā) is the probability of the hoop landing entirely within one square.

What is the volume of the largest cube that fits inside a sphere of unit volume ?

3/(4*pi)

too hasty. (2)/(pi*sq root 3). But that seems far too small,so another error has crept in. Simple trigonometry letting me down. A day and half spent with young children in the house has clearly addled my brain  

NAJB posted:

too hasty. (2)/(pi*sq root 3). But that seems far too small,so another error has crept in. Simple trigonometry letting me down. A day and half spent with young children in the house has clearly addled my brain  

If you evaluate your answer and it is close to 0.368, you might surprise yourself

Cheers

Don

NAJB posted:

3/(4*pi)

Yes, that one was a bit hasty............

OK I'll bite 50rpm since the first and last wheels are the same. 

Don Atkinson posted:

Don Atkinson posted:

A fair coin is tossed four times in succession. What is the probability of getting exactly two heads and two tails ?

I'll bring this one and the next one forward before they get lost.

This one came directly out of the O-Level maths book so really can’t be that difficult. But it seems to keep getting overlooked !

16 outcomes from 4 coin tosses, to get 2H & 2 T, HHTT, HTHT, HTTH, THTH, THHT, TTHH, so 6/16 I think.

Eoink, you think right.

I wrote it down as 3/8, but nothing wrong with 6/16 ..........