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Non-prime series

Posted by Chris under MathsChallenge (No Respond)

Show that no member of the infinite series:
10001, 100010001, 1000100010001, 10001000100010001, … is prime.

Warning: this one might be hard. e.g. the 18th term’s smallest prime factor is 722817036322379041

Alice and Bob’s card game

Posted by Chris under Logic (8 Responds)

Two perfect logicians, Alice and Bob, play a game with 2n blank cards. The cards are numbered with random positive integers and laid out in a row. Alice goes first. She takes a card from either end of the row. The value of that card is added to her score. Bob then takes a card from either end of the remaining row and adds its value to his score. This continues until all the cards have been taken.

Show that Alice can always match or beat Bob’s score.

I’ve been driving in my car, it’s not quite a Jaguar…

Posted by Karl Sharman under Tom (4 Responds)

I went for a drive, and curiosly discovered that my car (Kevin the Capri) only has three speeds. It travels downhill at 72 mph (miles per hour), on the level at 63 mph, and uphill at only 56 mph Kevin the Capri takes 4 hours to travel from Tomsville in Tomshire to Tomerton in Tomerset. The return trip takes 4 hours and 40 minutes.
How far is it from Tomerton to Tomsville?

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Not a UK Billion.

Posted by Karl Sharman under Tom (6 Responds)

Find the smallest natural number greater than 1 US billion (10^9) that has exactly 1000 positive divisors. (The term divisor includes 1 and the number itself. So, for example, 9 has three positive divisors.) Please note that this is the improper US billion, not the correct UK billion – 10^12 ;-)

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I haven’t lost my marbles, I keep them in a cone…

Posted by Karl Sharman under Tom (4 Responds)

Five marbles of various sizes are placed in a conical funnel. Each marble is in contact with the adjacent marble(s). Also, each marble is in contact all around the funnel wall.
The smallest marble has a radius of 8mm. The largest marble has a radius of 18mm. What is the radius of the middle marble?

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Anyone play 52 Pick-up?

Posted by Karl Sharman under Tom (10 Responds)

This question was asked today, and the first person I thought of was Chris. I have no answer yet, and the fact that probabilities come into it, means I never will….
A standard pack of cards is thrown into the air in such a way that each card, independently, is equally likely to land face up or face down. The total value of the cards which landed face up is then calculated. (Card values are assigned as follows: Ace=1, 2=2, … , 10=10, Jack=11, Queen=12, King=13. There are no jokers.)
What is the probability that the total value is divisible by 13?

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What’s the Time, Mister Wolf?

Posted by Karl Sharman under Tom (2 Responds)

The minute hand of a clock is twice as long as the hour hand. At what time, between 00:00 and when the hands are next aligned (just after 01:05), is the distance between the tips of the hands increasing at its greatest rate?

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Wicked

Posted by Chris under MathsChallenge (11 Responds)

A set of positive integers is defined to be wicked if it contains no three consecutive integers. Count the sets with 0, 1 or 2 elements as wicked sets. Find the number of wicked subsets of the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. If it helps, define good sets as those with at least 3 consecutive digits.

It’s a long way to Luxor

Posted by Karl Sharman under Tom (11 Responds)

The towns of Cairo, Ptolemy, and Luxor are equidistant from each other. If a car is three miles from Cairo and four miles from Ptolemy, what is the maximum possible distance of the car from Luxor? Assume the land is flat

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Duck Shoot

Posted by Chris under Logic (5 Responds)

Ten perfect shooters simultaneously and independently fire at ten ducks. On average, how many ducks are killed?