An ant starts out, facing North, on an infinite chessboard full of white squares. Every time the ant reaches a white square, the square turns black, the ant turns 90 degrees to the right, and it moves one square forward. Every time the ant reaches a black square, the square turns white and the ant turns 90 degrees to the left, and it moves one square forward.
What happens after many moves? Where does the ant end up going?
Given a triangular array of dots of N rows, these questions ask whether you can cover all of them with blocks that cover 3 dots at a time ("tribones") and do not overlap.
For which N can you do this if the tribones cover three dots in a triangle?
For which N can you do this if the tribones cover three dots in a row?
You have a 5x5 grid with the numbers 1-25 and five clues:
Within each row and column, numbers are in increasing order.
Every row contains at least one odd number. Every column contains at least one even number.
No even numbers are in the top row. No odd numbers are adjacent to the number 10.
Column 4 has the most primes of any column. Row 4 has the fewest primes of any row.
All perfect squares are on the same diagonal.
Can you fill out the grid?
Hundreds of math/coding problems of all levels (including the question below).
In Nim, 2 players take turns removing stones from one of three heaps. On their turn, players can take any number of stones from one heap, but cannot remove stones from multiple heaps.
The first player to remove all the stones wins.
Suppose that the three heaps have n stones, 2n stones, and 3n stones and that players play optimally. For how many values n < 2^k does player 2 win?
As you may know, P. Pritzker donated $100 million to Harvard's economics department in 2021. Rumor had it that two other wealthy benefactors were also considering donating $100 million dollars. However, many believe that their strategies weren't pure.
Why?
(Hint: Look for the other titles where they previously put up capital.)