Programming Praxis

We’re going to try something different today. We often have interview questions here, but always of the type that require writing a program. Today, we will have one of the brain-teaser type of interview question:

How many raindrops fall on the planet every year?

Your task is to estimate the answer to the question posed above. When you are finished, you are welcome to read a suggested solution or to discuss your solution in the comments below.

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Yesterday was Alan Turing’s birthday, so today we will write a program for a turing machine.

The Busy Beaver is a turing machine that performs the maximum work for a given configuration of machine; the concept was invented by Tibor Radó in his 1962 paper On Non-Computable Functions. We won’t look at the underlying theory, although it is fascinating if you have the time. Instead, we’ll be content to implement the first few busy beavers. Here are the two-symbol busy beavers for one, two, three and four states, from Wikipedia; the two symbols are 0 and 1, the states are letters A through D plus the halting state H, and the moves L and R are left and right:

Your task is to implement the four busy beavers. 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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The birthday paradox, which we studied in a previous exercise, states that in any group of 23 people there is a 50% chance that two of them share a birthday. The BBC recently published an article that shows 16 of the 32 World Cup teams, each consisting of 23 players, have shared birthdays, thus demonstrating the paradox precisely. Today’s exercise asks you to recreate their calculation.

You can obtain the same listing of player birthdays that the BBC used from FIFA. Another source is the player rosters at WikiPedia.

Your task is to demonstrate that the 2014 World Cup rosters honor the birthday paradox. 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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Beware: today’s exercise sounds simple but is actually quite complex if you don’t look at it properly.

Your task is to write a function that takes three points in the x,y plane and determines if they are collinear; be sure to handle vertical lines and horizontal lines properly. 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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Over at /r/math, Darksteve writes of “A problem I came up with, and haven’t been able to solve for many years:”

I would like to present a mathematical problem that I came up with some years ago, and haven’t yet found a solution for:

If you take all the numbers that contain the digits 1 to 9 exactly once, and you write down the prime factorization of all those numbers, which one has the smallest biggest prime factor?

To illustrate what I mean, the number 879456123 contains the prime factors 3 7 13 and 491; making 491 this numbers biggest prime factor.

The number 213456789 contains 3 7 13 and 197 as factors, making 197 the biggest prime factor. Out of all the numbers I’ve tried, 213456789 has the smallest biggest prime factor.

Many other number have much bigger biggest prime factors, like 576492813 which contains 3 13 19 and 13649.

But I have not found a way to actually solve this problem, as I am lacking the programming skills, or the algebraic knowledge required. I would therefore greatly appreciate anyone capable of solving this.

Your task is to solve Darksteve’s problem. 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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I heard today’s exercise as an interview question — you have five minutes to solve this task, while I watch — but it could equally be a homework problem:

Write a function to return true/false after looking at a string. Examples of strings that pass:

{}, [], (), a(b)c, abc[d], a(b)c{d[e]}

Examples of strings that don’t pass:

{], (], a(b]c, abc[d}, a(b)c{d[e}]

Your task is to write the function 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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I’m not sure if this is a homework exercise or an interview questions, but it’s a fun little exercise to get your creative juices flowing on a Friday morning:

Given a string and a character, remove from the string all single occurrences of the character, while retaining all multiple consecutive instances of the character. For instance, given string “XabXXcdX”, removing singleton Xs leaves the string “abXXcd”.

Your task is to write the program given above; be sure to properly test your work. 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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We studied the problem of converting between integers and roman numerals in a previous exercise. We’ll do it again today because it’s a fun exercise, it appears frequently in lists of interview questions, we have an improved algorithm, and it lets us highlight a useful piece of the Standard Prelude.

Your task is to write functions that convert an integer to its equivalent in roman numerals and vice versa. 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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