You have three weights, each with an integer value, you can measure food parcels weighing between 1kg and 13kg. 
What are the value of the weights?

OK guys we haven’t posted to this in over ten months.   I will make one more attempt because I feel that the comments of Chris Grayling about bed and breakfasts need addressing (coming within the hour).

However there are a few things that we need to sort out if this “keep in touch blog” is ever going to take off.  The first is the incredibly annoying bots which keep posting thousands of comments (which have to be hand checked) and so effectively disable comments on the blog.  The other thing which would be good is an automatic email to a sign up-able list every time someone makes a new post.  Because, let’s be honest, we’re not going to keep checking a blog which is only posted to once every ten months.

Have posted this in: “George’s Problems” but really it is a problem for those who know a bit more about it.

Rescue trucks will be along shortly.  Will the dog be saved?

(Make reasonable estimates for frictional co-efficient, car weight and man’s strength)

These puzzles were inspired by a squash match I watched earlier today.  It was a best of five match and the Magdalene player was winning; two games to one.  The opposition player put most of his remaining “energy” into winning the fourth game.  In the event he put so much into winning that game that he was knackered by the time he got to starting the fifth (which the Magdalene Player won easily).  The question occured to me as to how hard each player should be trying, how much of the kitchen sink to throw into the match and how much to keep in reserve for the possible fifth game (?).

A more well defined problem is this:

Two players are both given £100.00 to use in wining a contest which is conducted as follows:  In the first round each player must pay as much or as little of his money into a box (unseen by his opponent) as he wishes to.  The boxes are then examined by the umpire who awards a point to whoever paid the most money (ties are scored as half a point each).  With whatever money each opponent has left a second round is conducted in the same way.  Finally a third round is conducted in which both players will place all their remaining money into a third and final box. The winner of the third round is decided in the same way.

Imagine you are playing this game.

Clearly, since there is symmetry in the game, there can be no strategy that guarantees a win for one or other player.

But is there one which guarantees at least a draw?

Assuming the answer to this question is no (which I feel would not be hard to prove), is there an optimal (most likely probabilistic rather than deterministic) strategy that maximises the chance of winning the contest? (Strictly I would want to maximise the expected outcome of the contest for the player employing the strategy. A drawn overall outcome would count as half a win)

Clearly if the above strategy exists it will have an expected outcome of 0.5 or better against all other strategys including itself (against which it would score, on average, 0.5).

Problem:

Find such a strategy or prove that one does not exist!

For all I know these two problems might be among the hardest in Mathematics or they might have relatively simple solutions. (This is especially likely if the answer is that such a strategy can be proved not to exist which I think is likely to be the answer).

Finally, and I think more easily, if you could cheat by bringing extra money into the game how much extra money would you need, with a good strategy, to guarantee that you win the contest?

For this one can you think of a good bound and accompanying strategy?

show

A problem from a book I read when I was small:

I have a draw with ten red socks and twenty black socks in it.  how many socks do I have to take out (in the dark) in order to be guaranteed a matching pair of socks.

Answer: show

You are driving in a car at a constant speed.

On your left side is a ‘drop off’, the ground is 18-20 inches below the level you are travelling on, and on your right side is a fire engine travelling at the same speed as you.

In front of you is a galloping horse, which is the same size as your car and you cannot overtake it.

Behind you is a galloping zebra. Both the horse and zebra are also travelling at the same speed as you.

What must you do to safely get out of this highly dangerous situation?

Answer: show