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.

Puzzle of the Week #557 - Palindromic Numbers

I learned the interesting fact the other day that any natural number, however large, can be written as the sum of at most three palindromic numbers. For instance, 7369 = 7227+131+11.

 Can you write the ten digit number 7,275,640,031 as the sum of three palindromic numbers?

 Single digit numbers count as palindromes (eg, 7) but numbers with leading zeroes do not (eg 0330).