Hector's Puzzled

Suppose you have precisely enough blocks to build a staircase with $n$ steps, each $1$ block deep and tall. Given that you must build rows and columns of the blocks up from the floor and out from the wall, how many distinct ways are there to build this staircase? fivethirtyeight. [Solution]

Assuming a deck is randomly shuffled before every game, how many games of War would you expect to play until you had a game that lasted just 26 turns? fivethirtyeight. [Solution]

For the problem statement, see the post at fivethirtyeight. [Solution]

The tortoise and the hare are about to begin a 10-mile race along a “stretch” of road. The tortoise is driving a car that travels 60 miles per hour, while the hare is driving a car that travels 75 miles per hour. (For the purposes of this problem, assume that both cars accelerate from 0 miles per hour to their cruising speed instantaneously.) The hare does a quick mental calculation and realizes if it waits until two minutes have passed, they’ll cross the finish line at the exact same moment. And so, when the race begins, the tortoise drives off while the hare patiently waits. But one minute into the race, after the tortoise has driven 1 mile, something extraordinary happens. The road turns out to be magical and instantaneously stretches by 10 miles! As a result of this stretching, the tortoise is now 2 miles ahead of the hare, who remains at the starting line. At the end of every subsequent minute, the road stretches by 10 miles. With this in mind, the hare does some more mental math. How long after the race has begun should the hare wait so that both the tortoise and the hare will cross the finish line at the same exact moment? (fivethirtyeight) [Solution]

What is the smallest $N$ such that the set of the first $N$ cubes can be partitioned into $K$ subsets whose sums are equal? [Solution]