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Posted by Chris under MathsChallenge (13 Responds)

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?

Golden Needle in a Haystack

Posted by DP under MathsChallenge (7 Responds)

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.

Untwinned twins

Posted by Chris under MathsChallenge (2 Responds)

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’s cash

Posted by Chris under MathsChallenge (7 Responds)

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?

The vexatious vault value veracity validation problem

Posted by Chris under Logic, Mathemagic, MathsChallenge (12 Responds)

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!

Can You Bet Better?

Posted by Karl Sharman under Tom (4 Responds)

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?


Easy Maths Primer

Posted by Karl Sharman under Tom (3 Responds)

The prime 41, can be written as the sum of six consecutive primes:

41 = 2 + 3 + 5 + 7 + 11 + 13

This is the longest sum of consecutive primes that adds to a prime below one-hundred.

The longest sum of consecutive primes below one-thousand that adds to a prime, contains 21 terms, and is equal to 953.

Which prime, below one-million, can be written as the sum of the most consecutive primes?


A Roll of The Dice

Posted by Karl Sharman under Tom (6 Responds)

Suppose n fair 6-sided dice are rolled simultaneously. What is the expected value of the score on the highest valued die?


Lowest Value of n is 404?

Posted by Karl Sharman under Tom (11 Responds)

Where the numbers 2n and 5n (where n is a positive integer) start with the same digit, what is the lowest possible value of n? The numbers are written in decimal notation, with no leading zeroes. I am going to get flak for this…. but, I have broad shoulders!


If we are doing dice – try some different dice!

Posted by Karl Sharman under Tom (9 Responds)

With a standard pair of six sided dice, there is one way of obtaining a 2, two ways of obtaining a 3, and so on, up to one way of obtaining a 12. Find all other pairs of six-sided dice such that:
a. The set of dots on each die is not the standard {1,2,3,4,5,6}.
b. Each face has at least one dot.
c. The number of ways of obtaining each sum is the same as for the standard dice.

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