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#338062 lamp algorithm
Posted by BMAD on 15 April 2014  05:47 AM
N lamps are set in a circle, and for each integer M you have a tool that can toggle the state (on/off) of any set of M consecutive lamps.
Find a possible N which satisfies the following statements:
The sum of its digits is less than 10.
By applying the tool for M=105 several times, we can toggle a single lamp.
If we remove one lamp and start from a random initial setting for the remaining N1 lamps, the probability that there exists a way to apply the tool for M=32 several times and switch all the lamps off is positive and less than 0.0011%.
#338061 foreign country currency
Posted by BMAD on 15 April 2014  05:30 AM
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#338049 Probability: equal sequences
Posted by BMAD on 14 April 2014  02:58 AM
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#338043 more fun with coins
Posted by BMAD on 11 April 2014  07:51 PM
Suppose John tosses a coin 250 times and Eric tosses a coin 251 times, what is the probability that john's coin has more heads than Eric? What is the probability that Eric has more heads than John?
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#338003 Broken keyboard
Posted by BMAD on 06 April 2014  02:00 AM
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#337994 alternative to Pachinko
Posted by BMAD on 04 April 2014  08:47 PM
This is a simple game like Pachinko and in a similar vein as my "coin with a conscience" problem.
You insert money into the machine at one metal ball per 10cents. The balls drop one at a time onto discs. The first ball drops and lands on a disc and then rolls and lands on the next disc, and then falls on two more (for a total of 4 discs). If after landing on the fourth disc, the ball rolls off to the left, you get 20cents. if the ball rolls off to the right, you earn nothing. After inserting your money, you choose whether the 1st ball should aim for the left part of the first disc or the right part of the first disc (and the ball will always go down that particular side). Once a ball goes down a particular side, the rest of the discs are rigged to bias the odds the ball falls the other way by a change of 5%.
For examplewith 1 ball
 aim for right side, disc 1(goes down right side), disc2 52.5% (50% * 1.05 = 52.5) likely to fall down left, disc 3 52.5% likely to fall down left, disc 4 52.5% likely to fall down left
 ball reaches disc2 say it falls down the left side, 49.875% likely to fall down the right side of disc 3, 49.875% likely to fall down the right side of disc 4
 ball reaches disc3, say it falls down the left side of disc3, 52.36875% chance it will fall down the right of disc4
 say the ball rolls off the right side of disc4....darn, i lost 10cents.
Can you make money playing this game in the long run? If so, how should you play it? If not, why not?
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#337988 Minor pieces on a chessboard
Posted by BMAD on 03 April 2014  12:53 PM
Consider a an infinite chessboard.
How many squares can a knight reach after precisely n moves?
How about a bishop?
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#337982 in terms of 5
Posted by BMAD on 02 April 2014  02:38 AM
Instead of the more traditional approaches of solving with respect to a variable, solve it with respect to 5. Prove your answer is correct by using substitution with your answer into the number 6.
E.g. since 5 + 5/5 = 6 then
Solution for 5 + solution for 5 ÷ 5
Should equal solution for 5 + 1
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#337917 hidden race order
Posted by BMAD on 28 March 2014  03:22 PM
The 400 metre dash sprinting event will be held at a field track with 5 lanes. 25 athletes will be participating in total, of which, obviously, only 5 can be running together at a time. Define the minimum number of dashes required to determine the 3 fastest athlets of all, so that they are awarded the gold, silver and bronze medal. Which athletes will be running in each dash?
We assume that each athlete performs exactly the same in each dash. The results of the event will be determined by the relative classification of the athletes and not by their exact times. We only need to determine the 3 fastest athletes and not to follow the exact procedure which usually is followed at such events. (Obviously we cannot use a stopwatch).
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#337899 A coin with a conscience
Posted by BMAD on 25 March 2014  02:46 PM
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#337898 Buying more elevators
Posted by BMAD on 25 March 2014  02:38 PM
There is a building with 50 floors. Every morning 100 guest, 2 on each floor, come to use a single elevator. The elevator has a max capacity of 10 users. The elevator begins on the 50th floor and travels down. Once the elevator achieves maximum capacity, the elevator travels only one additional floor. At this floor, everyone on the elevator exits and those waiting (who have not yet ridden in the elevator) get on the elevator, then the elevator continues with only these new riders. Everyone above the 1st floor desires to make it to the first floor while those on the first floor wish to make it to the fiftieth floor.
How many additional elevators should this building have (assume all elevators begin on the fiftieth floor)? Provide a quantifiable justification to support your answer (e.g. defend why 2 elevators are better than any other case, or 3, or 4 or...so on.)
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#337897 A deterministic probability game
Posted by BMAD on 25 March 2014  02:11 PM
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#337841 Santa Claus's speed
Posted by BMAD on 19 March 2014  03:57 AM
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#337840 gun with unlimited bullets
Posted by BMAD on 19 March 2014  03:48 AM
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#337790 Majority Rules
Posted by BMAD on 14 March 2014  05:45 PM
In an presidential election between Bipa and Viktor, the winning candidate Bipa received n+k votes, whereas Viktor has received n votes. (n and k are positive integers.) If ballots are counted in a random order, what is the probability that Bipa's accumulating count will always lead his opponent's, and why?
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