Puzzles
16 December
Some numbers can be written as the sum of two or more consecutive positive integers, for example:
$$7=3+4$$
$$18=5+6+7$$
Some numbers (for example 4) cannot be written as the sum of two or more consecutive positive integers.
What is the smallest three-digit number that cannot be written as the sum of two or more consecutive positive integers?
7 December
There are 8 sets (including the empty set) that contain numbers from 1 to 4 that don't include any consecutive integers:
How many sets (including the empty set) are there that contain numbers from 1 to 14 that don't include any consecutive integers?
2 December
What is the smallest number that is a multiple of 1, 2, 3, 4, 5, 6, 7, and 8?
5 December
Today's number is the number of ways that 35 can be written as the sum of distinct numbers, with none of the numbers in the sum being divisible by 9.
Clarification: By "numbers", I mean (strictly) positive integers. The sum of the same numbers in a different order is counted as the same sum: eg. 1+34 and 34+1 are not different sums.
The trivial sum consisting of just the number 35 counts as a sum.