Curious Numbers

June 28, 2016

Today’s exercise is in the style of Project Euler, so the rules are that your solution must not use more than a minute of computer time and that you can’t peek at the solution until you have the answer yourself.

Some numbers have the curious property that when they are squared, the number appears in the least-significant digits of the product. For instance, 6252 = 390625, and 71063762 = 50543227109376.

What is the sum of all numbers less than 1020 that have this curious property?

Your task is to find the sum described 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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John Cook is a mathematician and programmer who runs a fascinating blog that I frequent.

Cook recently had an article about the prime factors of telephone numbers. He explained that, for 10-digit telephone numbers as used in the United States, the average number of distinct prime factors is 3.232 and the distribution is between 1 and 5 distinct prime factors about 73% of the time.

Your task is to write a function that determines the number of distinct prime factors of a number, and use that function to explore the distribution of the number of distinct prime factors in a range of telephone 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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Two Interview Questions

June 21, 2016

I like to read a web site called Career Cup, both to enjoy solving some of the programming exercise given there and to find exercise for Programming Praxis. As I write this exercise, here are the two most recent exercises on Career Cup:

  • Given a function rand2 that returns 0 or 1 with equal probability, write a function rand3 that returns 0, 1 or 2 with equal probability, using only rand2 as a source of random numbers.
  • Given a set of characters and a dictionary of words, find the shortest word in the dictionary that contains all of the characters in the set. In case of a tie, return all the words of the same (shortest) length.

Your task is to write the two programs described 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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Here is Sakamoto’s algorithm for calculating the day of the week, taken from the comment that introduces the code:

Jan 1st 1 AD is a Monday in Gregorian calendar.
So Jan 0th 1 AD is a Sunday [It does not exist technically].

Every 4 years we have a leap year. But xy00 cannot be a leap unless xy divides 4 with reminder 0.

y/4 – y/100 + y/400 : this gives the number of leap years from 1AD to the given year. As each year has 365 days (divdes 7 with reminder 1), unless it is a leap year or the date is in Jan or Feb, the day of a given date changes by 1 each year. In other case it increases by 2.

y -= m So y + y/4 – y/100 + y/400 gives the day of Jan 0th (Dec 31st of prev year) of the year. (This gives the reminder with 7 of the number of days passed before the given year began.)

Array t: Number of days passed before the month ‘m+1’ begins.

So t[m-1]+d is the number of days passed in year ‘y’ upto the given date.

(y + y/4 – y/100 + y/400 + t[m-1] + d) % 7 is reminder of the number of days from Jan 0 1AD to the given date which will be the day (0=Sunday,6=Saturday).

int dow(int y, int m, int d) {
static int t[] = {0, 3, 2, 5, 0, 3, 5, 1, 4, 6, 2, 4};
y -= m < 3;
return (y + y/4 - y/100 + y/400 + t[m-1] + d) % 7;
}

Another description is given here.

Your task is to write a program that implements the day-of-week algorithm shown 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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Today’s exercise is in two parts, first a commonly-seen programming exercise and then a variant on it; the origin of the exercise is certainly someone’s homework, but since school is out for the year it doesn’t matter that we do the exercise today.

First, write a program that, given an array of integers in unsorted order, finds the single duplicate number in the array. For instance, given the input [1,2,3,1,4], the correct output is 4.

Second, write a program that, given an array of integers in unsorted order, finds all of the multiple duplicate numbers in the array. For instance, given the input [1,2,3,1,2,4,1], the correct output is [1,2,1].

Your task is to write the two programs that find duplicates in an array. 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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Linear Regression

June 10, 2016

Linear regression is a widely-used statistical technique for relating two sets of variables, traditionally called x and y; the goal is to find the line-of-best-fit, y = m x + b, that most closely relates the two sets. The formulas for computing the line of best fit are:

m = (n × Σxy − Σx × Σy) ÷ (n × Σx2 − (Σx)2)

b = (Σym × Σx) ÷ n

You can find those formulas in any statistics textbook. As an example, given the sets of variables

x    y
60   3.1
61   3.6
62   3.8
63   4.0
65   4.1

the line of best fit is y = 0.1878 x − 7.9635, and the estimated value of the missing x = 64 is 4.06.

Your task is to write a program that calculates the slope m and intercept b for two sets of variables x and y. 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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Christian Goldbach (1690-1764) was a Prussian mathematician and contemporary of Euler. One of the most famous unproven conjectures in number theory is known as Goldbach’s Conjecture, which states that every even number greater than two is the sum of two prime numbers; for example, 28 = 5 + 23. We studied Goldbach’s Conjecture in a previous exercise.

Although it is not as well known, Goldbach made another conjecture as follows: Every odd number greater than two is the sum of a prime number and twice a square; for instance, 27 = 19 + 2 * (2 ** 2). (The conjecture is sometimes stated as every odd composite number is the sum of a prime number and twice a square, since it is trivially true with 0 as the square root for all prime numbers; alternately, it is sometimes limited so that the number being squared must be positive, in which case there are some odd primes that can be so expressed.) Sadly, it is easy to find a counter-example that disproves Goldbach’s other conjecture.

Your task is to write a program that finds the smallest number that disproves Goldbach’s other conjecture. 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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A Dozen Lines Of Code

June 3, 2016

Today’s exercise demonstrates that it is sometimes possible to do a lot with a little.

Your task is to write some interesting and useful program in no more than a dozen lines of code. 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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