Find a formula for each of the following:
a) The number of walks with n steps starting from the origin, where each step is of the form (0,1) or (1,0).
b) The number of walks with n steps starting from the origin, where each step is of the form (0,1), (0,-1) or (1,0), and the walk does not intersect itself.
Alright, let’s revive this place !
How many real solutions are there to the equation
I wanted to repost the polyominoes question:
“Show that any polyomino with at 4n-2 squares in it can be properly cut into 2 polyominoes, each with at least n squares in them”
The hint I’ll give is to use induction.
As mentioned by someone, this is impossible for a polyomino with 4n-3 squares in it. This can be seen by taking a “+” shaped polyomino with 1 square in the centre, and a row of n-1 squares coming out of each side.
Here’s something that the more computer-inclined people can simulate.
Trodgor the burninator is burning down a village consisting of 90 cottages. At time t = 0 an angry peasant arises from each cottage, and every 8 minutes (480 seconds) thereafter another angry peasant spontaneously generates from each non-burned cottage. It takes Trodgor 5 seconds to either burn a peasant or to burn a cottage, but Trodgor cannot begin burning cottages until all the peasants around him have been burned. How many seconds does it take Trodgor to burn down the entire village?
These logic puzzles are fun.
A math professor stands up in front of a room containing 100 very smart math students and says, “Each of you has to write down an integer between 0 and 100, inclusive, to guess ‘two-thirds of the average of all the responses.’ Each student who guesses the highest integer that is not higher than two-thirds of the average of all responses will receive a prize.” If among all the students it is common knowledge that everyone will write down the best response, and there is no communication between students, what single integer should each of the 100 students write down?
Show that a polyomino with at least 4n-2 squares in it can be divided into two polyominoes with at least n squares in each.
More from Harvard, cuz I’m really enjoying these problems.
A student at Harvard named Kevin
Was counting his stones by 11
He messed up n times
And instead counted 9s
And wound up at 2007
How many values of n could make this limerick true?
This one is a little less imaginative, but people seem to have had fun with the last one.