Ruth-Aaron Pairs

August 11, 2017

We have an exercise today from the realm of recreational mathematics, based on a video from Numberphile. The two numbers 714 = 2 × 3 × 7 × 17 and 715 = 5 × 11 × 13 have, between them, the first seven prime numbers as their factors (baseball fans will understand the name of this exercise, others will have to watch the video). So the first exercise is to find other pairs of consecutive numbers that have as their factors all and only the first n primes, for some n.

Carl Pomerance, the speaker in the Numberphile video, credits one of his students for first noticing that the sums of the prime factors of 714 and 715 are equal: 2 + 3 + 7 + 17 = 5 + 11 + 13 = 29. So the second exercise is to find other pairs of consecutive numbers whose factors sum to the same number. This exercise can be divided into two parts: numbers where repeating factors are added in to the sum and numbers where only the distinct factors are added in to the sum.

Your task is to complete the three exercises; they will make much more sense if you watch the video. 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 a programming interview question from a programmer who very obviously didn’t get the job:

Given a C-style null-terminated string, move all the spaces in the string to the beginning of the string, in place, in one pass.

The candidate claimed it couldn’t be done, and apparently got into an argument with the interviewer, and took to the internet after the interview to vindicate himself about the “unfair question.”

Your task is to either write the requested program, or demonstrate that it cannot be done. 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 task is simple:

You are given a file containing one name per line in the format last name comma first name, optionally followed by another comma and a numeral (like Sr., or Jr., or IV), and are convert it to a file containing the names, one name per line, in the format first name, last name and numeral with no commas. You may assume the input is correctly formatted, with optional spaces after each of the two commas.

Your task is to convert a file of names to the correct format. When you are finished, you are welcome to read a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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Find Target In A Matrix

August 1, 2017

Consider a two-dimensional matrix containing integer entries in which all rows and all columns are sorted in ascending order; for example:

 1 12 43 87
 9 25 47 88
17 38 48 92
45 49 74 95

Your task is to write a program that takes a matrix as describe above, and a target integer, and determines if the target integer is present in the matrix. 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’ve written about binary search trees on several occasions, most recently here. Binary search trees are a fruitful source of exercises, and we have another one today:

Write a program that takes a binary search tree and a given node of the tree and returns a new binary search tree with the given node at its root, changing as few nodes within the tree as necessary.

Your task is to write a program to reroot a binary search tree. 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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Ladder Range

July 25, 2017

Once a year, or thereabouts, I pick up Jon Bentley’s book Programming Pearls and read a random chapter; even though I’ve read it before, re-reading always makes the material seem fresh. Today’s exercise is Exercise 7 from Chapter 4 about binary search:

A colleague faced the following problem in a program to draw lines on a bit-mapped display. An array of n pairs of reals (ai, bi) defined the n lines y = mi x +
bi. The lines were ordered in the x-interval [0, 1] in the sense that yi < yi+1 for all values of i between 0 and n − 2 and all values of x in [0, 1].

[ Here Bentley has a picture of a ladder with rungs at various angles to the horizontal. We won’t reproduce it here; get the book if you want to see it. ]

Less formally, the lines don’t touch in the vertical slab. Given a point (x, y), where 0 ≤ x ≤ 1, he wanted to determine the two lines that bracket the point. How could he solve the problem quickly?

Your task is to write a program to solve Bentley’s exercise. 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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Length Of A Cycle

July 21, 2017

Our exercise today is about finding cycles in a linked list. We’ve seen algorithms due to Robert Floyd (the tortoise-and-hare algorithm) and Richard Brent (the power-of-two algorithm) in previous exercises, but in those cases all we were interested in doing was in finding a cycle if it existed. In today’s exercise we want to find the length of the cycle and the list item that begins the cycle (the list item that has two inward pointers).

Your task is to modify both Floyd’s and Brent’s algorithms to find the length and location of a cycle. 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 an interview question:

You are given a binary search tree in which all keys in the left child of a node are less than or equal to the key of the current node and all keys in the right child of a node are greater than or equal to the key of the current node. Find the most common key in the binary search tree. You may use O(n) time and O(1) space.

Your task is to write a program to find the most common node in a binary search tree, subject to the given constraints. 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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Nuts And Bolts

July 14, 2017

Today’s exercise is an interview question from Microsoft:

You are given two bags, one containing bolts and the other containing nuts, and you need to find the biggest bolt.. You may compare bolts to nuts, to see which is larger, but you may not compare bolts to bolts or nuts to nuts. Write a program to find the biggest bolt.

Your task is to write a program to find the biggest bolt. 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 recently purchasd a Lenovo TAB2 A10 tablet computer (who thinks up these horrible names?) with 2GB RAM and a gorgeous 1920 × 1280 screen; the tablet has been on the market for about two years, so it’s no longer cutting edge, but the software is upt to date and the beautiful screen makes up for any deficiency. I bought it as a poor-man’s laptop, intending to carry it with me pretty much everywhere. I’m writing this exercise on my new tablet.

One of the programs I installed from the Google Play Store is GNUroot, which despite its name doesn’t root the tablet; it installs a Unix-like system within the sandbox of a normal Android application. It provides a console that looks like an ordinary Unix console. The sole user is root, with no password; a VNC server is provided if you want to use a graphics screen. The root directory has the normal Unix file structure, with /bin, /usr, /lib, /var, /etc, /home and all the others, and all the normal Unix utilities are present, including apt-get, which lets you install most of the GNU programs.

A simple apt-get install guile-2.0 gave me Guile, the GNU Scheme interpreter, which I’ve been playing with for the last few days. Guile is aggressively R5RS, with lots of extensions and libraries that are inconsistent with R6RS and R7RS; for instance, the module system is completely different. My first impression is good, even though the arguments to (sort list-or-vector lt?) are in the wrong order, and I’ll be exploring the library for the next few days. My .guile initialization file appears on the next page.

So there is no exercise today. You might wish to tell us about your computing environment or ask questions about GNUroot in the comments below.

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