## Leading Digits Of Powers Of 2

### June 23, 2017

John D. Cook, a programmer who writes about mathematics (he would probably describe himself as a mathematician who writes about programming) recently wrote about the distribution of the leading digits of the powers of 2, observing that they follow Benford’s Law, which we studied in a previous exercise.

Your task is to write a program that demonstrates that the distribution of the leading digits of the powers of 2 follows Benford’s Law. 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.

## Mangarevan Counting

### June 20, 2017

Six hundred years ago, the people of the French Polynesian island of Mangareva developed a mixed-radix counting system that combined binary and decimal elements to count from 1 to 799. They had no zero. The digits 1 through 9 had their normal decimal value. Digits K, P, T and V had values 10, 20, 40 and 80, respectively, so they increased in a binary progression. A number *N* was represented as *N* = *n*V + T + P + K + *m*, where *n* and *m* were digits; note that T, P and K did not have modifiers. Thus, 73 is represented as TPK3, 219 is represented as 2VTK9, and 799 is represented as 9VTPK9 in Mangarevan. You might enjoy this article in *Nature* and this article in the *Proceedings of the National Academy of Sciences*. Arithmetic is interesting: 1VPK9 + 1 = 1VT, and 3VPK3 + 2VTK9 = 6VK2.

Your task is to write programs that translate to and from Mangarevan counting 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.

## A Scheme Idiom

### June 16, 2017

While I was reading some Scheme code recently, I discovered a delightful little Scheme idiom that could simplify some coding tasks. It looks like this:

> (define (make-accum n) (case-lambda (() n) ((m) (set! n (+ n m)) n))) > (define a (make-accum 20)) > a #<procedure> > (a) 20 > (a 10) 30 > (a) 30

Variable *a* is a accumulator; `define`

it to set its initial value, fetch its current value by calling it as a function, and increment it by calling it with a value. This works because function `make-accum`

returns a function, defined by `case-lambda`

, with a semantics that varies based on its arity: with no arguments, the function returns the value stored in the closure, and with a single argument, it increments the value stored in the closure and returns the new value. The actual value is stored inside the function closure so it is only available through the defined interface, making it “safer” in some sense. And the concept works for other data types than accumulators, as the solution page will show.

Your task is to describe a useful idiom in your favorite programming language. 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.

## Climb To A Prime

### June 13, 2017

The British mathematician John Horton Conway, famous for inventing the Game of Life cellular automaton, made this conjecture:

Select a number, then compute its prime factors, with multiplicity; for instance, 90 = 2 × 3

^{2}× 5. Then “bring down” the exponent and write the resulting digits, forming a new number; for instance, the exponent of 2 in the above factorization is brought down, forming the number 2325. Repeat the process with the new number, and again, and so on; for instance, starting from 90, the chain is 90, 2325, 35231, 72719, where the chain terminates. I conjecture that the process will eventually terminate with a prime number.

At his YouTube channel, Numberphile revealed that the conjecture is false. The number 13532385396179 = 13 × 53^{2} × 3853 × 96179, so at each step it replaces itself, resulting in an infinite loop that will never reach a prime, thus disproving the conjecture. The discoverer of that number, James Davis, is entitled to a $1000 prize from Conway.

Your task is to write a program that calculates the climb to a prime for a given input number. 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.

## Generating Large Random Primes

### June 9, 2017

We studied Pocklington’s Criterion, which lets us quickly find large random primes, in a previous exercise. That algorithm generates a *certified* prime — a number that is proven to be prime — rather than a *probable* prime according to some pseudoprimality test.

Even though it’s not hard to generate a certified large prime, most cryptographic applications accept probable primes, primarily because it is much faster to generate a probable prime than a certified prime. Wikipedia explains the algorithm:

For the large primes used in cryptography, it is usual to use a modified form of sieving: a randomly chosen range of odd numbers of the desired size is sieved against a number of relatively small primes (typically all primes less than 65,000). The remaining candidate primes are tested in random order with a standard probabilistic primality test such as the Baillie-PSW primality test or the Miller-Rabin primality test for probable primes.

Your task is to write a program that implements the Wikipedia algorithm for generating large random probable primes. 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.

## Matrix Rotation

### June 6, 2017

We have a two-part exercise today, based on a Microsoft interview question.

First, write a program to rotate an *m* × *n* matrix 90° to the right, as shown below; your solution should touch each matrix element only once:

a b c d e f m j g d a A = g h i rot(A) = n k h e b j k l o l i f c m n o

Second, write a program to rotate a square matrix with *n* rows and columns *in-place*. where the source and target matrices are the same matrix and there is no intermediate matrix (be sure your solution works for both even and odd *n*):

a b c d e u p k f a f g h i j v q l g b B = k l m n o rot(B) = w r m h c p q r s t x s n i d u v w x y y t o j e

Your task is to write the two programs that rotate matrices. 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.

## Second Largest Item

### June 2, 2017

Today’s exercise is only slightly tricky:

Write a program to find the second largest item in an array. Use the least possible number of comparisons.

Your task is to write a program to find the second largest item in an array using the least possible number of comparisons. 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.

## Sieving A Polynomial

### May 30, 2017

I watch Stack Overflow and Reddit for questions about prime numbers. Most of the questions are basic, many are confused, a few are interesting, and some are just bizarre. A recent question falls in the latter category:

I need to make a list of the prime factors with multiplicity of all the numbers up to 300,001 that are equivalent to 1 modulo 4. My strategy is to make a list of the numbers 1 modulo 4, then calculate the factors of each. I don’t know how to calculate factors, but I know it’s complicated, so I figure somebody has already built a factoring program and put it on the web. Does anybody know a web API that I can call to determine the factors of a number?

For instance, if we limit the numbers to 50, the desired output is:

5 (5) 9 (3 3) 13 (13) 17 (17) 21 (3 7) 25 (5 5) 29 (29) 33 (3 11) 37 (37) 41 (41) 45 (3 3 5) 49 (7 7)

Calling a web API 75,000 times is a spectacularly wrong way to build the list, but I pointed the questioner to Wolfram|Alpha. Long-time readers of this blog know how to solve the problem directly, which we will do in today’s exercise.

Your task is to write a program to factor all the numbers up to 300,001 that are equivalent to 1 modulo 4. 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.

## Hill Cipher

### May 26, 2017

The Hill Cipher was invented by Lester Hill in 1929. A block cipher based on modular matrix multiplication, it was rather advanced for its time. The diffusion inherent in the algorithm makes it difficult to break manually, though it is easy to work back to the key if you have some known plaintext.

The Hill Cipher uses arithmetic, so you will need to convert the alphabetic plaintext to numbers using a polybius square, a straddling checkboard, or the simple expedient of mapping A to 0, B to 1, and so on, until Z is 25, which we will adopt here. Our alphabet has 26 characters, which can be a problem since 26 = 2 × 13 is composite; the alphabet size becomes the modulus of the cipher system, and some keys won’t work, though it is easy to find keys that do. If you’re worried, add characters to the alphabet until it has a prime number of elements; for instance, you might choose a 29-character alphabet with the letters A to Z, a space character, a period, and a slash for a digit shift.

Let’s look at an example. We will send the message PROGRAMMINGPRAXIS with blocksize 3. The plaintext is padded with a Z, forming a 6 × 3 matrix:

P R O 15 17 14 G R A 6 17 0 P = M M I = 12 12 8 N G P 13 6 15 R A X 17 0 23 I S Z 8 18 25

We must choose a key that is invertible; that is, a key that has an inverse (some textbooks call such keys non-singular). For a normal matrix to be invertible, the discriminant must be non-zero. Since we are using modular arithmetic instead of normal integer arithmetic, we have the additional requirement that the determinant and modulus must be coprime. If the first key you choose doesn’t work, try another. The key that we choose, and its mod 26 inverse, are:

G Y B 6 24 1 8 5 10 I F K K = N Q K = 13 16 10 inverse = 21 8 21 = V I V U R P 20 17 15 21 12 8 V M I

Encryption is performed by modular matrix multiplication, with C = P × K (mod 26):

15 17 14 19 12 5 T M F 6 17 0 6 24 1 23 0 23 X A U C = 12 12 8 * 13 16 10 = 24 18 18 = Y S S 13 6 15 20 17 15 14 13 12 O N M 17 0 23 16 19 24 Q T Y 8 18 25 2 21 17 C V R

So the ciphertext is TMFXAUYSSONMQTYCVR.

Decryption is also performed by modular matrix multiplication, with P = C × K_{inv} (mod 26):

19 12 5 15 17 14 P R O 23 0 23 8 5 10 6 17 0 G R A P = 24 18 18 * 21 8 21 = 12 12 8 = M M I 14 13 12 21 12 8 13 6 15 N G P 16 19 24 17 0 23 R A X 2 21 17 8 18 25 I S Z

Your task is to write a program that performs encryption and decryption using the Hill Cipher. 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.

## Matrix Determinant And Inverse

### May 23, 2017

Today’s exercise is preliminary to the exercise we will have later this week. You are to write programs that calculate the determinant and inverse of a matrix. I won’t go into the math involved behind the matrix arithmetic, as there are many sources on the internet that know far more about the process than I. Google for “matrix determinant” or “matrix inverse”; I used YouTube for my instruction.

Your task is to write programs that calculate the determinant and inverse of a 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.