Puzzle of the Week # 579 - Trip Counter

True story: when I bought my Dacia Sandero a few months ago I immediately reset the trip counter to zero. The overall mileage of the car was 55664. I noticed that I was only 0.1 miles into ownership when the overall mileage changed to 55665, so I can surmise it was actually on 55664.9 when I bought it, even though it only displays full miles. The trip counter displays the mileage to an accuracy of 0.1 miles, and if you don’t manually reset it can count as far as 9999.9.

At some point, will the overall mileage and the trip counter mileage appear the same, with the only difference the decimal point in the trip counter mileage? If so, what will that mileage be?

Puzzle of the Week #578 - Prime Bridge

The first nine prime numbers (2,3,5,7,11,13,17,19,23) are trying to get across a bridge. They only have one car, which they are each capable of driving, and which can hold a maximum of three of them. The time the car takes to cross the bridge is whatever the highest prime number in the car for that crossing, for instance if the 3, 7 and 17 are in the car it would take 17 minutes to cross.

What is the shortest time in which all the prime numbers can cross the bridge?

Puzzle of the Week #576 - Overlapping Chessboards

I take a pair of 10x10 chessboards. Initially I overlay them exactly on top of one another and find the point 4 squares along and 2 squares down and drive a nail through both boards. The I rotate the top board around the nail until the bottom left corner of the upper board coincides with the bottom edge of the lower board.

I take a second pair of 10x10 chessboards and do exactly the same except that this time the nail is driven through at a point 3 squares along and 1 square down.

The question is: in which scenario is the area of overlap between the upper and lower chessboards greatest?

Puzzle of the Week #568 - Dice

I have a standard six sided dice marked 1,2,3,4,5,6. Since the probability of rolling a 6 is 1/6, if I keep rolling till I get a 6 it would take on average 6 rolls.

How many rolls on average would it take to roll a six if I only consider sequences of rolls that DON’T contain any 1s?

Puzzle of the Week #567 - Basketball Tournament

Seven teams took part in a basketball league tournament. Each possible pair of teams either played each other once or twice.

After the end of the tournament the coach of the Nuneaton Predators asked the other six coaches how many games their teams had each played, and was surprised to receive six different answers.

How many games did the Nuneaton Predators play?

Puzzle of the Week #562 - Zig Zag Sequence

I’ve constructed a sequence thusly:

I’ve started with the natural numbers 1,2,3,4,5,6,7,8,9,10 etc.

I’ve kept the first number as it is, then reversed the order of the next two, then reversed the order of the subsequent three, then reversed the order of the next four etc:

1,3,2,6,5,4,10,9,8,7,15,14,13,12,11,21,etc

 

Your task is to construct a formula that will instantly give you the nth term in this crazy sequence.