快猫短视频

#28 Dice differences

set by Peter Rowlett

We play a game starting with a counter in the centre of a row of squares. Every turn, we each roll a six-sided die. If the difference between the values rolled is 2 or less, the counter moves one place to the left. If the difference is more than 2, it moves one place to the right. I win if the counter reaches the left-hand end of the row and you win if it gets to the right.

Am I offering you a fair game or does one direction have an advantage?

The game is adjusted so that if the difference is 2 or less, the counter still moves one place to the left, but if the difference is more than 2, it moves two places to the right. Under these rules, what is the smallest number of rolls after which the counter could return to its starting position?

Is this adjusted game fair?

Solution next week

#27 Dancing decimals

Solution

1/11 = 0.090909鈥�, with 鈥�09鈥� repeating forever. 2/11 = 0.181818鈥� and 3/11 = 0.272727鈥� . The two repeating digits are 9 times the fraction鈥檚 numerator, so 7/11= 0.636363鈥� and 10/11 = 0.909090鈥� (this repeat occurs because 9 脳 11 = 99). 1/7 = 0.142857鈥� (and 142857 脳 7 = 999,999, hence a six-digit repeating string). 2/7 = 0.285714鈥�, the same six digits looping but starting from 2. Putting the other options in numerical order, we predict that 3/7 = 0.428571鈥�, 4/7 = 0.571428鈥�, 5/7 = 0.714285鈥� and 6/7 = 0.857142. The thirteenths have two different strings, based on 1/13 = 0.076923鈥� and 2/13 = 0.153846鈥� . Finding all the ways to rotate them and sorting them in numerical order, we predict that 3/13 = 0.230769鈥� and so on.