Generalize to an MxN chess board: if both M and N are odd, under what conditions can the board be tiled if one square is removed; if M or N is even and two squares are removed, under what conditions can the board be tiled.
Rob:
"I just picked two integers greater than one and their sum is less than 100."Adam:
"Adam, their sum is ..." (he whispers it to Adam).
"Rohit, their product is..." (he whispers it to Rohit).
"Rohit, we don't know the numbers."Rohit:
"Now I do."Adam:
"Me too".
What were the numbers?
- The piles contain 13,11,6 coins. Do you want to move first?
- The piles contain 13,11,22 coins. Do you want to move first?
- In a sealed room is a transparent box and an opaque box. The transparent box contains $100. The opaque box is currently empty. You go in and survey the scene.
- The rules of the game are as follows. You will eventually be given a choice. Either you can choose the opaque box only, or you can choose both boxes. I will consult my machine. If my machine predicts that you will take only the opaque box then I will insert $1000000 into it. If it predicts that you will take both boxes, I will do nothing.
- So you now leave the room and I consult my machine. I do whatever I have to do and pick up my machine and leave the room. You enter and now have 1 hour to make your choice.
- What do you do? - Remember that the machine can essentially predict you with 100% accuracy.
- The problem is that you do not see all the applicants at once and you have to immediately notify the current applicant if you are hiring her or else s/he will just settle for some job say at Express or something, who knows.
- What is your best strategy?
Magdon's 1/e theorem for dating - what an application!
- Russian roulette is played as follows. Into a 6-shooter pistol is placed one bullet. The barrel is then rapidly rotated and comes to a stop. Now 2 players take it in turns to place the gun to their head and shoot. Eventually the game ends with one player dead. Do you want to shoot first or second? Generalize to K-shooter russian roulette played by M players and compute the probability that you stay alive if you follow the optimal strategy.
- A fair coin has probability of turning up heads (p) given by p=0.5. You and another player take turns to toss the coin. Whoever tosses the first head wins $1. Do you want to toss first or second. Generalize to M players. If you choose the optimal action (given p and M), what is your probability of winning (as a function of p and M).
- Aren't the two problems essentially the same? The second seems to be the equivalent to the first for very large K, in fact the limit as K goes to infinity. So how come the answers are different?
Balls now in your court!
(Bonus: What speed would you suspect the phenomenon to pop up at - you'll be surprised)
- What about if Dick now came along. Is there a way to divide with 3 cuts so that each receives what they believe is at least 1/4 of the cake. What about 4 cuts? 5 cuts?...
- Suppose now each kid is happy if they each believe they have more than their fair share. Can this be done with 3 kids?
- Suppose now that each kid is happy only if they believe they have more cake than anyone else. Can this be done with 3 kids?
- Can you generalize the problem and your algorithms?
- In how many ways can 37 cents be made up using 1 and 2 cent stamps. Generalize to n cents using 1 and k cent stamps.
- In how many ways can 37 cents be made up using 1, 2 and 3 cent stamps. Generalize to n cents using 1, 2 and 3 cent stamps.
- Show that using only 4 and 5 cent stamps, you can come up with any postage greater than 11 cents. Thus, it suffices to stock up on only 4 and 5 cent stamps. For example the current postage is 37c which is eight 4c stamps and and one 5c stamp.
- Show that using only 6 and 7 cent stamps, any total greater than 29c can be made.
- The maharaja wants to be able to drink wine in a month, what is the minimum number of tasters the maharaja should be willing to put out of commission for at least a month in order to be able to drink wine safely in a month?
- The maharaja wants to throw an orgy in a month using all the uncontaminated amphoras. What is the minimum number of tasters that the maharaja must be willing to put out of commission (for a month) in order to be able to have this orgy? For example, a solution is to use 100 tasters, one on each amphora. One can do better though.
- A container with a small pebble is floating in a bucket of water. The pebble is taken out and dropped into the water. The pebble sinks to the bottom of the bucket. Does the level of the water in the bucket rise or fall?
- A cube of ice is also floating in the bucket. The ice subsequently melts. Does the level of the water in the bucket rise or fall?
Leaving the Grounds:
- What if the bulb were controlled by 3 switches on an equilateral triangle?
- What if the bulb were controlled by 4 switches on a square?
- can your algorithm be generalized...?
If instead of $1 I said if you fail you owe me $X, what is the maximum value of X for which you will take on this bet?
What if only one person (either a perpetual liar or perpetual truth teller) was standing at the fork. What question would you ask.
- What is the optimal strategy for each player? With best play, who is most likely to win?
- Suppose the triplets were chosen in secret? What then would be the optimal strategy?
- What would be the optimal strategy against a randomly selected triplet?
What about if player A has N>1 more coins than B?
What is the value of the sum |a1 - b1| + |a2 - b2| + ... + |an - bn|?
|
Two concentric circles are as pictured. The chord in the larger one which is tangent to the smaller one has length 100. What is the area of the ring between the two circles? |
Show [x+y]&ge[x]+[y].
- Show that the product of n consecutive positive integers is divisible by n!.
- If a,b,c,... are positive integers with a+b+c+...&le n then n!/(a!b!c!...) is an integer.
![\begin{displaymath}
1=\sqrt[3]{\frac{13\sqrt{3}}{36}+\frac{5}{8}}-
\sqrt[3]{\fra...
...}{8}}
\qquad
2=\sqrt[3]{6\sqrt{3}+10}-
\sqrt[3]{6\sqrt{3}-10}
\end{displaymath}](img1.png)
![\begin{displaymath}
1=\sqrt[3]{\frac{1}{8}+\frac{c^2}{2}+\frac{c(4c^2+9)\sqrt{3}...
...qrt[3]{\frac{1}{8}+\frac{c^2}{2}-\frac{c(4c^2+9)\sqrt{3}}{36}}
\end{displaymath}](img3.png)
to get my solutions
or to tell me of an extremely ingenious and short solution
that you have found.