A Fistful of Brain Teasers

An infinitely large floor is covered with hexagonal tiles side L. Different coloured tiles are used such that no two adjacent tiles are the same colour.

A thin hoop diameter D is thrown onto the floor at random.

What is the probability of the hoop laying across more than one colour of tile ?

You want to be an astronaut ?  (as an alternative to popeye's what DO you do ?)

• Box of matches
• Handheld GPS receiver
• 15 metres nylon rope
• Three spare batteries for spacesuit
• Parachute silk
• Three spare carbon-dioxide removal canisters for spacesuit
• One case of dehydrated food
• Three spare oxygen tanks for spacesuit
• Stellar map
• Self-inflating life raft
• Magnetic compass
• 20 litres of water – can be administered through a special drinking port in spacesuit
• Two handheld mirrors
• First-aid kit including medical tape, scissors etc
• Solar-powered FM receiver-transmitter

You and two crew members have crash-landed on the moon 50 km from your intended rendezvous with the Lunar Base. Survival depends on reaching that Lunar Base. Your spacecraft and most of its contents have been destroyed but the most critical items from the above list of undamaged items must now be selected for your journey.

The good new is that in the lunar gravity, you can travel at 5 km/hr. The bad news is that your built-in spacesuit consumables will only last for 8 hours.

Group the above items in order of importance/usefulness for reaching the lunar base. Put them into three groups viz – Essential; Useful; Not much use.

PS it's only a game, there is no risk that you, or any other "middle manager" will be on the next rocket to the moon

Ian G. posted:

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

Apologies for the delayed response, Ian.

You’re right. And I think it doesn’t matter what’s in between, but if the first and last wheels are the same size, they rotate at the same speed.

If so, it follows that all wheels of the same size, turn at the same speed.

How many squares are there on a chess board ?

Don Atkinson posted:

How many squares are there on a chess board ?

Depends on who's playing ?

Don Atkinson posted:

How many squares are there on a chess board ?

Lots.

Black or white?

202

8*8 +7*7 + ....1*1

Although the "squares on a chess board" also came out of he O-Level book, I have always seen it as a "Character Assessment" type of question rather than simply a maths question.

Sure, the question as presented is somewhat vague. and It is presented as a "Puzzle", so more for amusement than O-Level progress. Unlike the "proper" questions, there are no answers given at the back of the book for puzzles !

So, 10/10 for all the Character Assassination responses - well done  - and i'm sure there are plenty more !

As for IB's second attempt............... well it's along the right lines ...................

IB made a good attempt but .....

..... for the life of me I can't generate 202 from 8*8 + 7*7 + ...... + 1*1

Perhaps I should have checked either my summing or used a calculator... try 204!

Well done IB !

Eventually,

A starter for 10 for the hoop in a hexagon teaser above.

The method for the hexagon problem is the same as for the Hoop La once if that encourages anyone else to have a go. 

Probability of hoop spanning more than 1 colour in the hexagonal tiling is 

Ian, check your answer for (say) D = 1 and L = 2

oops 

Note to self:  1 don't try and do more in your head than you can and 2 - sense check your answers!

Ian G. posted:

oops 

Note to self:  1 don't try and do more in your head than you can and 2 - sense check your answers!

Ah !

Much better and....... much neater into the bargin !! Brilliant

BTW, the formula only holds good for values of "D" less than or equal to L?3

Ian, my formula is different to yours, but they both generate the same probabilities.

And they are both limited to values of D less than or equal to L?3

Going back to the hoop and hexagon. Let l (lower case L) be the side of the small hexagon, the the required probability is area of the

P = (area large hexagon - area small hexagon)/area large haxagon.

as the hexagon is just 6 equilateral triangles then each area is proportional to the square of the side ( actually?3L²/8)

P = (L² - l²)/L² = 1 - l²/L²

the height (H) of the equilateral triangle side L 

H = ?3L/2

the height (h) of the small hexagon is

h = H - D/2 > 0 

D < 2H 

D < ?3L

P can be restated in terms of H, h as

P = 1 - h²/H²

or

P = 1 - ((H - D/2)/H)²

P = 1 - (1 - D/?3L)²

As Ian posted.

Nice explanation Phil.

Don Atkinson posted:

Nice explanation Phil.

I thought it might help. Your drawing was a vital part. Thanks for setting these challenges.

Phil

Ok, a couple of straight forward probability tasters.

The first one inspired by those tubes of fruit pastilles...........

A box contains 2 red, 3 yellow and 5 green sweets. One is taken out at random and eaten. A second sweet is then taken out.

1. If the 1st sweet was green, what is the probability that the second sweet is also green ?

2. If the first sweet was not red, what is the probability that the 2nd sweet is red ?

The second one incorporates those proverbial fair dice..........(six sided cubes with sides numbered 1 to 6)

Two dice are thrown together.

1. What is the most likely score ?

2. what is the chance of getting this score three times in successive throws ?