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Alan and Bob’s cash

Posted by Chris under MathsChallenge (4 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 saya 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 (11 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 (3 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?

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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?

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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?

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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!

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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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Another die rolling game

Posted by Chris under Logic, MathsChallenge (10 Responds)

Sue and Bob take turns rolling a fair 6-sided die. Once either person rolls a 6 the game is over. Sue rolls first, if she doesn’t roll a 6, Bob rolls the die, if he doesn’t roll a 6, Sue rolls again. They continue taking turns until one of them rolls a 6.

If Bob rolls a 6 before Sue, what is the probability that he did it on his second roll?

500 Triangles

Posted by Karl Sharman under Tom (5 Responds)

A sequence of triangle numbers is generated by adding the natural numbers. So the 7th triangle number would be 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28.
The first 7 terms would be: 1, 3, 6, 10, 15, 21, 28

Let us list the factors of the first seven triangle numbers:
1: 1
3: 1,3
6: 1,2,3,6
10: 1,2,5,10
15: 1,3,5,15
21: 1,3,7,21
28: 1,2,4,7,14,28

We can see that 28 is the first triangle number to have over five divisors.

What is the value of the first triangle number to have over five hundred divisors?

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Time Trial on Trust

Posted by Karl Sharman under Tom (9 Responds)

I have a selection of 4 fairly easy puzzles. You can do these in your head. Time yourself, and, if you dare – publish your time along with the answers. Read the questions carefully, and your time starts……. NOW!

1. A large water tank has two inlet pipes (a large one and a small one) and one outlet pipe. It takes 2 hours to fill the tank with the large inlet pipe. On the other hand, it takes 5 hours to fill the tank with the small inlet pipe. The outlet pipe allows the full tank to be emptied in 7 hours. What fraction of the tank (initially empty) will be filled in 1.35 hours if all three pipes are in operation? Give your answer to two decimal places (e.g., 0.25, 0.5, or 0.75).

2. The son of a rich bullion merchant left home on the death of his father. All he had with him was a gold chain that consisted of 98 links. He rented a place in the city center with a shop at the lower level and an apartment at the upper level. He was required to pay every week one link of the gold chain as rent for the place. The landlady told him that she wanted one link of the gold chain at the end of one week, two gold links by the end of two weeks, three gold links by the end of three weeks and so on. The son realized that he had to cut the links of the gold chain to pay the weekly rent. If the son wished to rent the place for 98 weeks, what would be the minimum number of links he would need to cut?

3. A cylinder 48 cm high has a circumference of 16 cm. A string makes exactly 4 complete turns round the cylinder while its two ends touch the cylinder’s top and bottom. How long is the string in inches?

4. My Dad has a miniature Pyramid of Egypt. It is 3 inches in height. Dad was invited to display it at an exhibition. Dad felt it was too small and decided to build a scaled-up model of the Pyramid out of material whose density is (1 / 5) times the density of the material used for the miniature. He did a “back-of-the-envelope” calculation to check whether the model would be big enough. If the mass (or weight) of the miniature and the scaled-up model are to be the same, how many inches in height will be the scaled-up Pyramid?

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