Three recipents R(0), R(1) and R(2) each contain an integer volume v_{0}(0) ≥ v_{0}(1) ≥ v_{0}(2) ≥ 1. Each recipient is large enough to contain the combined volumes. You are allowed to transfer some liquid from one recipient to another, only if the receiving one doubles its volume. Show that there is always a way to empty out one recipient in finitely many steps.

Example: If the initial volumes are 17, 8, 5 the sequence of volumes could be

R(0) R(1) R(2)

17 8 5

17 3 10

17 6 7

17 12 1

16 12 2

14 12 4

14 8 8

14 16 0 => R(2) is finally empty.

A function f takes a positive integer and returns another one by moving the leftmost digit to the right.

For example f(12345) = 23451

What is the smallest strictly positive integer n such that f(n) = 1.5 n

In a test involving yes/no answers, the probability that the official answer is correct is t, the probability of getting the real correct answer is b for a boy and g for a girl. If the probability that a randomly chosen boy or girl of getting the official answer to a question is 1/2, then what is the ratio of boys to girls who took the test?

Find all solutions in positive integers a, b, c to the equation

a! b! = a! + b! + c!

916238457 is a nine-digit number that has the property that has the digits 1 to 9 exactly once and the digits 1 to 5 are in counting order but the digits 1 to 6 are not. How many such numbers are there?

I have a very large plot of land on which I have stacked bales of hay, each measuring 1m x 1m x 1m. They are stacked 10 bales high, and span a couple thousand wide and deep. To keep them out of the weather I have also placed a tarp on top that cannot be seen through and should not be walked on or cut open in any way.

While stacking the hay I also stored several boxes of gold for safe keeping, but did not mark down where. However, to make it a little easier for my future self I grouped 8 boxes together so that they occupy a 2m x 2m x 2m space.

I now need some of my gold, but cannot retrieve it myself. I am hiring YOU to help me, and in return you will get to keep 1 gold bar from each of the boxes you return to me.

I need the gold available in a week, so time is somewhat important. You may work whenever you like, and at whatever pace you like. I do not have a minimum or maximum requirement.

Some other information:

- You must enter through the ‘side’ as to not disturb my tarp or the ground.

- You should remove at least 2 bales high so you can walk through, more if you wish.

- Bales above those removed will remain in place (magic?) – same with the tarp. [I suppose in theory you could remove all adjacent bales and 1 might ‘float’ right in front of you.]

- Once a bale is removed, a face of the 5 adjacent (but not diagonal) bales/boxes are revealed. You may remove any of which you can see the face.

Thoroughness is partially important (finding ALL of the gold within the search area), but Efficiency will yield the most gold found per hay bale removed.

The question: What is the most Efficient method of finding my gold? (ex: remove all bales, remove in a checker-board pattern, make a long 5-bale-high tunnel, etc.)

This info might also be helpful: When I stacked the hay I placed them in 16 x 16 x 10 piles each day with one group of gold boxes ‘randomly’ placed within the pile. The next day I placed another 16 x 16 x 10 pile directly adjacent along with another group of gold boxes somewhere within, and so on.

A class has six pairs of twins. The teacher wishes to set up teams for a quiz, but doesn’t want to put any pair of twins in the same team.

1) In how many ways can they be split into two teams of six?

2) In how many ways can they be split into three teams of four?

Alan and Bob have a whole number of dollars. Alan says to Bob, “If you give me $3, I will have n times as much as you”. Bob says to Alan, “If you give me $n, I will have 3 times as much as you”.

If n is a positive integer, what are its possible values?

This problem was originally posted by Karl Sharman (with a less idiotic title).

Whilst I was away under the pretence of work, a nearby bank uncovered a plot to swap the gold in their vaults with counterfeits. It was determined that all the gold bars in three of the Bank’s seven vaults were replaced with counterfeits. The other four vaults were uncompromised. The plot was foiled through the poor math skills of the thieves: while the real gold bars weigh ten kilograms, the counterfeits all weighed nine kilograms.

I was asked to work out which was the real gold, and which was the fake. I, being really bad at maths, so Chris tells me decided to recruit your help.

Your mission, should you wish to accept it, is determining which vaults have real gold, and which are just gold-plated bars of platinum.

The Bank Director has made the following generous offer: If you can determine the counterfeits using just one weighing on a scale, you can keep one bar as a souvenir.

Here are the rules:

This is a scale, not a balance, but you can weigh as many bars together as you like.

Only one weighing!

The bars will be handled by professional guards, so you won’t have a chance to “feel” their weights.

Each vault contains several hundred bars.

The guards have requested that you try to keep the number of bars you need to a minimum.

How do you do it, and what is the minimum number of bars…?

Good luck!

Would you take this bet?

Your friend he’s going to pick two different random numbers from a distribution not known to you. He will then write the two numbers of two small pieces of paper and put each one in each of his enclosed hands. You then pick a hand and he will reveal the number in that hand.

You then place your money on which hand you think contains the larger number. He will match your money, betting on the other hand (regardless of whether or not you chose correctly — this is a precondition), the numbers will then be revealed and whoever was right keeps the sum of the money.

You only get one shot at this, and the bet is £1… If you can’t afford to lose it – don’t bet! The question is, is there a strategy to beat 50:50 odds and if so how?

Tags:

Maths