Fogcreek programmers Variation of Hat Puzzle

100 fogcreek programmers are lined up in a row by an assassin. the assassin puts red and blue hats on them. they can’t see their own hats, but they can see the hats of the people in front of them. the assassin starts in the back and says “what color is your hat?” the fogcreek programmer can only answer “red” or “blue.” the programmer is killed if he gives the wrong answer; then the assassin moves on to the next programmer. the programmers in front get to hear the answers of the programmers behind them, but not whether they live or die. they can consult and agree on a strategy before being lined up, but after being lined up and having the hats put on, they can’t communicate in any way other than those already specified. what strategy should they choose to maximize the number of programmers who are guaranteed to be saved?

Solution

this is a very difficult problem to solve during an interview (especially if you’ve already taxed the candidate’s brain). look for obvious solutions first, and the reasoning behind them and then try to lead them to the ultimate solution.

a logical answer could be all the programmers would just say “red” and that way about half of them would survive on average, assuming the hats were distributed randomly.

this is a good start and should naturally lead to having every other programmer say the color of the hat in front of them. the first programmer would say the color of the hat in front of him, then the next programmer would just say that color that was just said. so we can guarantee that half survive - the even numbered programmers (since the person behind them told them the answer). and potentially if the hats were distributed randomly some of the programmers would get lucky and the hat in front of them would be the same color as their own. so this strategy should save more than half, and on average 75% of them would live.

at this point, if the solution is not clear, the candidate may give answers like, “they could agree that if they said their hat color in a soft voice, it means the hat in front of them is the same color, and if they say it in a loud voice, it means the hat in front is a different color”. this is definitely good and on the correct track. another option is they could say “reeeeeeeeeeed” for x number of seconds, where x represented the distribution of hats where a hat was a bit in a binary number, (red = 1, blue = 0). another interesting answer. there are many others like these that “bend” the rules and come to a solution.

but the real solution acknowledges that the programmers can only say “red” or “blue” and cannot alter their voice in such a convincing way as to signal any information other than the word they said. a good way to get this point across, is simply to change the problem slightly by saying “the assassin gets to hear their plan before she puts the hats on, and so will try to thwart the plan however she can.”

so if they decide to all say “red”, she’ll put blue hats on all of them. if they decide to all say the color of the hat in front of them, she’ll alternate the hats on every head, guaranteeing half will die. even with the assassin hearing their plan, there is still a way to save almost everyone.

we know that the first person is never going to have any information about the color of their hat, so they cannot be guaranteed to survive. but, i’ll give you a hint to the solution: i can save every other person for sure.

solution: they agree that if the number of red hats that the back person can see is even, that programmer will say “red”. if they add up to an odd number, they will say “blue”. this way number 99 can look ahead and count the red hats. if they add up to an even number and number 100 said “red”, then 99 must be wearing a blue hat. if they add up to an even number and number 100 said “blue”, signalling an odd number of red hats, number 99 must also be wearing a red hat. number 98 knows that 99 said the correct hat, and so uses that information along with the 97 hats in front to figure out what color hat is on 98’s head.

sample:

100 99 98 97 96 95 94 ... facing ->
R B B R B R B ... -> 45 R and 48 B

this shows #100 wearing a red hat, 99 a blue, 98 a blue, 97 a red, 96 a blue, 95 a red, 94 a blue and 45 red hats - 48 blue hats on the people in front of them.

100 counts up the red hats: 47 total. so 100 says “blue”. the assassin kills 100. 99 counts up the red hats in front: 47. 100 said blue, so 100 saw an odd number. 99 sees an odd number, so 99 says “blue” and lives. 98 had counted 47 red hats, and 99 didn’t say “red” so thats still the total. 98 says “blue”. 97 counts up and finds 46 red hats. 99 and 98 didn’t say “red”, so his count is missing a red hat (its on his head, he realizes). he says “red”. 96 heard the “red” and now knows that there are an even number of “red” hats in front of 95. 96 sees 46, so he knows he has a “blue” hat. etc…

even if the assassin knows the plan, she can’t thwart it. she hears the plan, but she still has to put the hats on their heads. the plan doesn’t rely on any ordering of the hats, so the worst the assassin can do is to make sure #100 gets killed and thats the worst damage she can do.

Check if Given Date is Palindrom or Not

Problem: this year on October 2, 2001, the date in MMDDYYYY format will be a palindrome (same forwards as backwards).
10/02/2001
when was the last date that this occurred on? (see if you can do it in your head!)

Solution: we know the year has to be less than 2001 since we already have the palindrome for 10/02. it can’t be any year in 1900 because that would result in a day of 91. same for 1800 down to 1400. it could be a year in 1300 because that would be the 31st day. so whats the latest year in 1300 that would make a month? at first i thought it would be 1321, since that would give us the 12th month, but we have to remember that we want the maximum year in the 1300 century with a valid month, which would actually be 1390, since 09/31 is a valid date.

but of course, a question like this wouldn’t be complete without an aha factor. and of course, there are not 31 days in september, only 30. so we have to go back to august 08 which means the correct date would be 08/31/1380.

palindromes also offer another great string question.
write a function that tests for palindromes
bool isPalindrome( char* pStr )

if you start a pointer at the beginning and the end of the string and keep comparing characters while moving the pointers closer together, you can test if the string is the same forwards and backwards. notice that the pointers only have to travel to the middle, not all the way to the other end (to reduce redundancy).

bool isPalindrome( char* pStr )
{
if ( pStr == NULL )
return false;

char* pEnd = pStr;
while ( *pEnd != '\0' )
pEnd++;

pEnd--;

while(pEnd > pStr)
{
if ( *pEnd != *pStr )
return false;

pEnd--;
pStr++;
}

return true;
}

How can you get a fair coin toss if someone hands you a coin that is weighted to come up heads more often than tails?

Hint:Treat outcome TH as tails, HT as heads, and reflip when you get TT and HH.

Explaination

f a cheat has altered a coin to prefer one side over another (a biased coin), the coin can still be used for fair results by changing the game slightly. John von Neumann gave the following procedure:[1]

1. Toss the coin twice.
2. If the results match, start over, forgetting both results.
3. If the results differ, use the first result, forgetting the second.

The reason this process produces a fair result is that the probability of getting heads and then tails must be the same as the probability of getting tails and then heads, as the coin is not changing its bias between flips and the two flips are independent. By excluding the events of two heads and two tails by repeating the procedure, the coin flipper is left with the only two remaining outcomes having equivalent probability. This procedure only works if the tosses are paired properly; if part of a pair is reused in another pair, the fairness may be ruined.

Mathematically We Can Write
Lets define an event, E as tossing the biased coin twice. The possible outcomes with probabilities is as follows

P(h,h) = x^2
P(h,t) = x(1-x)
P(t,h) = x(1-x)
P(t,t) = (1-x)^2

The event h,t or t,h are equi-likely, without any bias we can call that if Event h,t occurs it means head, t,h means tails but if h,h or t,t occurs we repeat the experiment.

Try to find the expected number of coin toss that would be required to call heads or tails?

Let expected coin toss be e.

Probability that we get outcome in 1st event is 2x(1-x)
Total number of coin toss would be 2
Probability that we get no outcome in 1st event is [1 - 2x(1-x)]
Total number of coin toss would be 2 + e (we wasted 2 coin toss and still we expect e)

==> e = 2 * 2x(1-x) + (2+e) * [1 - 2x(1-x)]
==> e = 4x(1-x) + 2 - 4x(1-x) + e - 2ex(1-x)
==> e = 1/ [x(1-x)]

so for x = 1/2, e = 4
for x=2/3, e = 4.5
for x=1, e = INF (as expected, because all we get is chain of h h h h h)


What is the probability that there wont be any outcome in e coin tosses (expected outcomes)?

p = prob of e heads + prob of e tail
==> p = x^e + (1-x)^e

To find a bound on this p in polynomial terms can be done by using binomial expansion and Newtonian series is out of the scope of this blog. But some number crunching is.

For x = 1/2, e = 4, p = 0.125
For x = 2/3, e = 4.5, p = 0.168
For x = 3/4, e = 5.33, p = 0.216
For x = 0.99, e = 101, p = 0.362

This tells us that probability that outcome comes is quite good even for high coin biases.


How can we improve on the expected number of coin tosses?

In above method, we say that HT or TH terminates experiment. and continue the experiment a fresh when the outcome is HH or TT.

We can further combine outcome of two such events to increase the probability of outcome e.g. say HH TT => heads and TT HH means tails.


How much expected coin tosses are we doing in this case?

try?

Feel Free to Write Comment or Suggestion

Find The Number of Pages in Printing Book,if Tottal Number of Digits Printed is Given

There is a printed book. The page numbers are not printed. Now the printing of page numbers is being done separately. If the total number of digits printed is 1095 then how many pages does the book have?



There are 9 significant single digit numbers. Hence for the first 9 pages, 9x1 = 9 digits will be printed.

There are 90 significant double digit numbers. Hence for the next 90 pages, 90x2 = 180 digits will be printed. Total number of pages = 99 (90 plus 9), total number of digits is 189(9 plus 180) till now.

Subsequently, we have 900 significant triple digit numbers and 900x3 = 2700 digits. Since the total number of digits printed lies in the range [189, 2889], it suggests that all double digit numbers were printed and some triple digit numbers were printed. The upper limit is arrived at as 2700 plus 189.

Number of digits used to print triple digit numbers = 1095 -189 = 906.

Therefore, 906/3 = 302 triple digit numbers were used.

Therefore the total number of pages printed = 9 plus 90 plus 302 = 401.

That is how solution: 401

Feel Free to Comment or Suggesting New Approaches .