## Thue-Morse Sequence

### September 30, 2014

Mathematicians are strange people. As an example, I offer the Thue-Morse sequence, which is a binary sequence of 0’s and 1’s that never repeats, obtained by starting with a single 0 and successively appending the binary complement of the sequence. Thus, term 0 of the sequence is (0), term 1 of the sequence is (0 1), term 2 of the sequence is (0 1 1 0), term 3 of the sequence is (0 1 1 0 1 0 0 1), term 4 of the sequence is (0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0), and so on; to calculate term 3 from term 2, we started with term 2 (0 1 1 0), complemented each element of the term (1 0 0 1), then appended the two (0 1 1 0 1 0 0 1). That looks useless to me, but mathematicians have been excited by it since 1851, hence my conclusion that mathematicians are strange people.

Your task is to write a program that generates the nterm of the Thue-Morse sequence. 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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## Blum’s Mental Hash

### September 26, 2014

It is generally accepted wisdom that people should use different passwords for each of their online accounts. Unfortunately, few people do that because of the difficulty of remembering so many passwords.

Manuel Blum — he’s the guy in the middle of the Blum Blum Shub sandwich — suggests an algorithm that hashes a web site name into a password. Further, his algorithm is cryptographically secure, so no one else can determine your password, and can be worked mentally, without a computer or even pencil and paper.

Blum’s algorithm works in two steps. A function f maps letters to digits, and a permutation g transposes the digits. The two, taken together, form your personal key, and so must be kept secret.

The first step maps letters to digits; since there are more letters than digits, some of the digits are used more than once. If, for instance, f(a) = 8, f(b) = 3, and f(c) = 7, then the first step would map the input “abc” to the digits [8, 3, 7].

The second step uses the permutation g to calculate the output. Begin with the first and last digits, adding them mod 10; in the example, (8 + 7) % 10 = 5. If the permutation g = 0298736514, then the next digit after 5 is 1, so the first output digit is 1.

After that, each remaining input digit is the basis of an output digit. Calculate the sum of the next input digit and the previous output digit, mod 10, and take the next digit of the permutation, repeating for each remaining input digit. In the example, the second input digit and the first output digit are summed mod 10, (3 + 1) % 10 = 4, and the next digit in the permutation is 0 (wrapping around), so the second output digit is 0. In the same way, the third output digit takes the third input digit and the second output digit, sums them mod 10, and computes the permutation, so (7 + 0) % 10 = 7, which permutes to 3. So the final password produced from an input of “abc” is “103”.

Your task today is to implement Manuel Blum’s mental hashing algorithm for mapping a web site name to a password. 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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## Triangle Roll-Up

### September 23, 2014

We have a simple little exercise today: Given a list, write a triangle showing successive sets of sums of the pair-wise elements of the list. For instance, given the input 4, 7, 3, 6, 7, your program should write this output:

81
40 41
21 19 22
11 10 9 13
4 7 3 6 7

The original list is at the bottom of the triangle. The next row up has pair-wise sums of the elements of the list: 4 + 7 = 11, 7 + 3 = 10, 3 + 6 = 9, and 6 + 7 = 13. The next row up has pair-wise sums of those list elements: 11 + 10 = 21, 10 + 9 = 19, and 9 + 13 = 22. The next-to-top row has only two sums: 21 + 19 = 40 and 19 + 22 = 41. And finally the top row is the sum of those two numbers: 40 + 41 = 81.

Your task is to write a program to print the triangle roll-up of an input list. 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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## Diophantine Reciprocals

### September 19, 2014

Career Cup claims that Amazon asked this as an interview question; it is also Problem 108 at Project Euler:

In the following equation x, y and n are positive integers: 1 / x + 1 y = 1 / n. For n = 4 there are exactly three distinct solutions: 1/5 + 1/20 = 1/6 + 1/12 = 1/8 + 1/8 = 1/4. What is the least value of n for which the number of distinct solutions exceeds one thousand?

Your task is to solve Amazon’s question; you might also like to make a list of the x, y pairs that sum to a given n. 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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## Torn Numbers

### September 16, 2014

In 1917, Henry Ernest Dudeney published a book Amusements in Mathematics of arithmetic puzzles. Today’s exercise solves puzzle 113 from that book:

A number n is a torn number if it can be chopped into two parts which, when added together and squared, are equal to the original number. For instance, 88209 is a torn number because (88 + 209)2 = 2972 = 88209.

Your task is to write a program to find torn 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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## Levenshtein Distance

### September 12, 2014

In a previous exercise we computed the edit distance between two strings where the allowable operations were insert or delete characters; we made the calculation by determining the longest common subsequence. A different definition of edit distance allows substitutions as well as insertions and deletions, and is called the Levenshtein distance, since it was studied by Vladimir Levenshtein in 1965. For example, the edit distance between “hill” and “hilt” is 2 (delete the “l” and insert the “t”) but the Levenshtein distance is 1 (replace “l” by “t”).

The basic algorithm is recursive: if either string is empty, the Levenshtein distance is the length of the other string, otherwise it is the minimum of the Levenshtein distance computed by deleting the first character from the first string, by deleting the first character from the second string, or by deleting the first characters of each of the two strings (adding 1 if the two characters differ), plus the Levenshtein distance of the remaining substrings. That computation takes exponential time as it recomputes the same substring Levenshtein distances many times. A better algorithm uses dynamic programming to build up the substring distances, so they are always available as they are needed.

Your task is to write two functions to compute the Levenshtein distance between two strings. 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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## Drawing Diamonds

### September 9, 2014

Given a small positive integer n, write a function that draws a diamond, either filled or in outline as specified by the user. For instance, here are filled and outline diamonds for n = 5:

```    *              *
* *            * *
* * *          *   *
* * * *        *     *
* * * * *      *       *
* * * *        *     *
* * *          *   *
* *            * *
*              *
```

Note that there is a single space between asterisks in the filled version of the diamond.

Your task is to write a program that draws diamonds as 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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## Skyline Puzzle

### September 5, 2014

The Skyline Puzzle is a classic programming exercise; it draws a silhouette of a city skyline by blocking out portions of buildings that are masked by taller buildings. A city is a list of buildings specified as triples containing left edge, height, and right edge. For instance, the list of triples `(1 11 5) (2 6 7) (3 13 9) (12 7 16) (14 3 25) (19 18 22) (23 13 29) (24 4 28)` encodes the eight buildings shown at the left of the diagram, and the path `1 11 3 13 9 0 12 7 16 3 19 18 22 3 23 13 29 0` encodes the skyline shown at the right of the diagram, where the odd-indexed elements of the output are the x-coordinate of the skyline and the even-indexed elements of the output are the y-coordinate of the skyline. (It makes more sense to me that the output should look like `(1 11) (3 13) (9 0) (12 7) (16 3) (19 18) (22 3) (23 13) (29 0)` but that’s not the way the puzzle is ever specified.) Notice that the second (2 6 7) and eighth (24 4 28) buildings are not part of the skyline.

Your task is to write a program that takes a list of buildings and returns a skyline. 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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## Longest Increasing Subsequence

### September 2, 2014

A sequence is a list of integers (or any other ordered type, but we’ll use integers to keep things simple). A subsequence is any, possibly non-consecutive, list drawn from the parent sequence with items in the same order as the parent sequence. An increasing subsequence is a subsequence with all the items in increasing order. A longest increasing subsequence (there may be more than one with the same length) is an increasing subsequence of a parent sequence of the greatest possible length. For instance, the sequence (3 2 6 4 5 1) has longest increasing subsequences (2 4 5) and (3 4 5).

The algorithm to find the longest increasing subsequence is similar to the algorithm for patience sorting of a previous exercise, with a small modification. When dealing the cards, each time a card is placed on a pile, a back-pointer to the top card on the previous pile is placed along with the card. Then, when all the cards are dealt, the number of piles is the length of the longest increasing subsequence, and the longest increasing subsequence can be recovered by taking the top card from the last pile and following the back-pointers to previous piles.

For instance, with sequence (3 2 6 4 5 1) the cards are dealt with 3 and 2 on the first pile, 6 and 4 on the second pile, 5 on the third pile, and 1 on the first pile, so the longest increasing subsequence has length 3. The 5 on the third pile ends the longest increasing subsequence, it points to the 4 which was on the top of the second pile when 5 was added to the third pile, and 4 points to the 2 which was on the top of the first pile when 4 was added to the second pile; even though 1 was later added to the first pile, is wasn’t yet on the pile when 4 was added to the second pile, so it’s not part of the longest increasing subsequence.

Your task is to write a program to find the longest increasing subsequence of a sequence. 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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