The iron bar recalls an old-time method for calculating the median of a stream; for instance, consider finding the median of a list of a thousand occurrences of the numbers 1 to 100. Initialize the median to the first number in the list. Then, at each subsequent number, increase the median by 1 if the number is higher than the current median and decrease the median by 1 if the number is lower than the current median. At the end of the list, the approximate median calculated in this way ought not to significantly differ from the actual median. This is essentially the same calculation as that of the iron bar.

Your task is to write a program that calculates approximate medians in the manner of an iron bar. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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Toss a coin four times. Write down the percentage of heads on the flips coming immediately after heads. Repeat that process one million times. On average, what is that percentage?

The article claims that the correct percentage is 40%, not the 50% that you would expect from an unbiased coin, and that this is proof that the hot hand exists, even from such a random source as coin flips. If you’re interested, you can read the academic article that the *Wall Street Journal* account is based on, or see discussion of it at *Hacker News*.

Your task is to write a program to confirm the 40% figure. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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Your task is to write the program that creates a palindrome. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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As was known to the ancient Greeks, perfect numbers such as 6 and 28 are equal to the sum of their aliquot divisors (those divisors less than the number itself); for instance, 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14. Mathematicians refer to these as P_{2} numbers the sum of the divisors, including the number itself, is twice the number.

Multiply perfect numbers are numbers such that the sum of their divisors are some multiple of the number; perfect numbers have divisor-sums twice the number, triply perfect numbers P_{3} have divisor-sums thrice, the number, and so on.

Your task is to write a program that finds multiply perfect numbers. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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1) Given a list of integers, return a list in which all the duplicates have been removed. For instance, given the input list [3, 2, 4, 2, 7, 3, 5, 1, 3] return the output list [3, 2, 4, 7, 5, 1].

2) Given two lists of integers, each sorted in ascending order, return a list of the integers common to the two lists; the output list must also be in ascending order. For instance, give the input lists [2, 3, 5, 5, 6, 7, 8, 9] and [1, 2, 4, 5, 5, 7] return the output list [2, 5, 5, 7].

3) Given a positive integer, determine if it is a perfect cube. For instance, the integer 125 is a perfect cube, 5*5*5, but the integer 121 is not.

Your task is to write programs that solve the three homework problems. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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1) Write a recursive function to find the sum of the first

nodd integers. For instance, ifn= 2, the first two odd integers are 1 and 3, and their sum is 4.2) Write a function to count the frequence of characters in a string. For instance, given the string “hello” the function should return counts of 1, 1, 2 and 1 for h, e, l and o.

3) Write functions that encrypt and decrypt messages in a Caesar cipher. Input consists of upper-case letters and spaces; the “key” is an integer giving the number of letters to shift. For instance, the message ATTACK AT DAWN with a shift of 5 is enciphered as FYYFHP FY IFBS.

Your task is to solve the three homework exercises given above. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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`julian`

and `gregorian`

for performing date arithmetic. In today’s exercise we will extend those functions to provide arithmetic on times as well. The parameters provided to `julian`

will be year, month, day, hour, minute, and second, and the same parameters will be returned by `gregorian`

.
Your task is to write the new `julian`

and `gregorian`

functions. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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Your task is to write a program that takes an array of 8-bit integers (possibly but not necessarily in sort) and finds the median value in the array; you should find an algorithm that takes linear time and constant space. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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In the days before computers, challenges such as this were frequently issued by one mathematician or group of mathematicians to their competing colleagues; for instance, the French mathematicians might challenge the German mathematicians. Once the challenge was met, the solving group would issue a new challenge to the original challengers. It was considered un-gentlemanly to issue a challenge to which you did not have a solution, though Fermat was probably respected enough to get away with it. Times have changed.

Your task is to write programs that find solutions to the two challenges. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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`int("".join(reversed(str(n))))`

has got to be far more efficient than using your recursive function. Your function is clever, but clever isn’t always best.” He was wrong, of course: arithmetic on numbers is better than converting a number to a string, creating a generator to reverse the individual characters of the string, appending them one-by-one to a new string, and converting the new string back to a number, which I demonstrated to him in a timing experiment.
Your task is to find the best way to reverse the digits of a positive integer, in the language of your choice. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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