## Two-Base Palindromes

### October 21, 2014

I wanted to do this exercise as a follow-up to the earlier exercise on generating palindromes, but didn’t get around to it until now.

Your task is to write a program that generates a list of numbers that are palindromes in base 10 and base 8; for instance 1496941_{10} = 5553555_{8}. 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.

## Blackjack

### October 17, 2014

Blackjack is a casino game of chance, played by a player and a dealer. Both player and dealer are initially dealt two cards from a standard 52-card deck. If the player’s initial hand consists of an ace and a ten or face card, the player wins, unless the dealer also has an ace and a ten or face card, in which case the game is a tie. Otherwise, the player draws cards until he decides to stop; if at any time the sum of the pips on the cards (aces count either 1 or 11, face cards count 10) exceeds 21, the player is busted, and loses. Once the player is finished, the dealer draws cards until he has 17 or more pips, or goes bust, at which time the game ends. If neither has gone bust, the hand with the most pips wins. There are many variant rules, but we’ll keep things simple with the rules described above.

Your task is to simulate Blackjack and determine the winning percentage for the player. 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.

## Formatting Text, Again

### October 10, 2014

Today’s exercise is a continuation of the previous exercise, which filled text to the maximum column width. Today’s exercise is to add justification to the previous exercise, so that extra space at the end of line of text is distributed to the spaces between the words on the line, making each line end at the right margin.

Your task is to write a program that fills and justifies text. 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.

## Text Formatting

### October 7, 2014

Text formatting is a huge topic. Today’s exercise looks at a simple text formatter. Input to the formatter is a file of ascii text; the input is free-form, except that paragraphs are marked by blank lines (two successive newline). The formatter copies the file to its output, moving text from one line to the previous line to fill each line as much as possible. It is possible to specify the width of a line, but if none is given the width defaults to sixty characters.

Your task is to write a simple text formatter. 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.

## Magic 1089

### October 3, 2014

I’ve been very busy at work this week, so I only have time for a simple exercise today.

This is a simple puzzle in arithmetic. Take any three-digit number with digits in descending order; for instance, 532 is acceptable, but 481 is not. Reverse the digits of the original number, and subtract the reversal from the original. Then reverse that difference, and add the reversal of the difference to the difference. Write the result as output.

For instance, start with the number 532. Its reversal is 235, and the difference is 532 – 235 = 297. Reversing the difference gives 792, and 297 + 792 = 1089, which is the result.

Your task is to write a program that makes this calculation, and try it on several different starting numbers; you might enjoy working out the arithmetic behind the results that you see. 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.

## 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 *n*term 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.

## 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.

## 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.

## 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,yandnare positive integers: 1 /x+ 1y= 1 /n. Forn= 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 ofnfor 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.

## 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

nis 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}= 297^{2}= 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.