Fibonacci Search

May 12, 2015

An interesting variant on binary search is Fibonacci search. Invented by Jack Kiefer in 1953 to find the zeros of a function, and first applied to searching in an array by David Ferguson in 1960, its initial appeal was to improve locality when searching for a record on magnetic tape. It was later applied to searching on paged memory when it was expensive to read a segment of an array from disk, and it is now used to improve locality of cache memory; a good idea never goes away! Here is a description of Fibonacci search, taken from Wikipedia:

Let Fk represent the k-th Fibonacci number where Fk+2=Fk+1 + Fk for k>=0 and F0 = 0, F1 = 1. To test whether an item is in a list of n ordered numbers, proceed as follows:

1) Set k = m, where Fm is the smallest Fibonacci number greater than or equal to n.
2) If k = 0, halt and report failure.
3) Test item against entry in position Fk-1.
4) If match, halt and report success.
5) If item is less than entry Fk-1, discard entries from positions Fk-1 + 1 to n. Set k = k – 1 and go to 2.
6) If item is greater than entry Fk-1, discard entries from positions 1 to Fk-1. Renumber remaining entries from 1 to Fk-2, set k = k – 2 and go to 2.

Your task is to implement Fibonacci search. 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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Monkeys And Coconuts

May 8, 2015

We have today a famous puzzle:

Five sailors are shipwrecked on a desert island. They quickly determine that the only other inhabitant of the island is a monkey and that the only food is coconuts. They set about collecting as many coconuts as they can and put them all in a pile. By nightfall they are too tired to divide the harvest; so they agree to go to sleep and divvy up the coconuts the next morning.

During the night one sailor awakens, suspicious that the others might try to cheat him, and desides to take his portion then and there and not wait until morning. He divides the coconuts into five piles and finds there is one coconut left over, which he gives to the monkey. He hides one of the five piles, then puts the rest of the nuts together and returns to sleep. About an hour later a second sailor awakens with the same suspicions and does the same thing: He divides the coconuts into five piles, leaving one extra, which he gives to the monkey. Then he hides what he thinks is his share and goes back to sleep.

One after another the rest of the sailors do the same: they each take one fifth of the coconuts in the pile (after giving the extra one to the monkey) and then return to sleep.

When the sailors awaken the next morning they all notice the coconut pile is much smaller than it was the night before, but since each man is as guilty as the others, no one says anything. They divide the coconuts (for the sixth time), but this time there is no coconut left for the monkey.

How many coconuts were in the original pile?

Your task is to determine how many coconuts were in the original pile; first solve the problem for 5 sailors, then again for 6 sailors, and finally for 30 sailors. 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 saw this question on a beginning programmer’s forum a couple of weeks ago. There were several answers, some of them wrong. So we’ll do it right:

Given an angle expressed in degrees, minutes, and seconds, convert it to radians. Given an angle in radians, convert it to degrees, minutes and seconds.

Your task is to write programs that perform the two conversions. 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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Collatz Primes

May 1, 2015

Today’s exercise comes from the world of recreational mathematics; I found it at Stack Overflow:

The Collatz sequence starting at n continues with n / 2, if n is even, and 3 n + 1 if n is odd. For instance, the Collatz sequence that starts from 19 is 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. It is conjectured that all Collatz sequences eventually end at 1, but has never been proven. The Collatz sequence that starts from 19 contains 7 prime numbers: 19, 29, 11, 17, 13, 5 and 2. Find the smallest starting number for a Collatz sequence that contains 65 or more primes.

Your task is to find the requested Collatz 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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Identifying Anagrams

April 28, 2015

Two words are anagrams if they consist of the same letters, with the same number of occurrences, in a different order. For instance, DEPOSIT and DOPIEST are anagrams (aren’t you glad you know that), and OPTS, POTS, TOPS and STOP form an anagram class.

Your task is to write a program that takes two strings as input and determines whether or not they are anagrams; you may assume that the strings consist of only the letters A through Z in upper case. You must provide at least two different algorithms that work in fundamentally different ways. 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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Minimum Impossible Sum

April 24, 2015

We have today another from our inexhaustible list of interview questions:

Given a list of positive integers, find the smallest number that cannot be calculated as the sum of the integers in the list. For instance, given the integers 4, 13, 2, 3 and 1, the smallest number that cannot be calculated as the sum of the integers in the list is 11.

Your task is to write a program that solves the interview question. 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 simple little interview question:

Generate the pairs of cartesian coordinates within a square bounded by (1,1) and (n,n) ordered by their product in ascending order. For instance, when n is 3, the coordinates are (1,1), (1,2), (2,1), (1,3), (3,1), (2,2), (2,3), (3,2) and (3,3). Can you find a solution with time complexity better than O(n2)?

Your task is to write a program to solve the interview question. 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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Fermat’s Last Theorem, which dates to the seventeenth century states that there are no solutions in integers to the equation xn + yn = zn for n > 2; the Theorem was finally proved a few years ago by Andres Wiles. In the eighteenth century, Euler conjectured that for any n > 2, it would take at least n terms of the form xin to sum to an n th power. That conjecture held until the age of computers, in 1967, when Lander and Parkin found the counter-example 275 + 845 + 1105 + 1335 = 1445.

Your task is to write a program that finds counter-examples to Euler’s Conjecture. 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 mixed-radix arithmetic in a previous exercise. In today’s exercise we look at a different kind of non-standard positional notation: balanced ternary, which is a base-3 number system that uses -1, 0 and 1 as its “trits” rather than 0, 1 and 2. For instance, the number -47 is written as (-1 1 1 -1 1) in balanced ternary, which is equivalent to -34 + 33 + 32 – 31 + 30. No separate sign is needed when using balanced notation; the sign of the leading trit is the sign of the whole number.

Arithmetic on balanced ternary numbers is done using the grade-school algorithms. Addition is done right-to-left with a carry; it is easy and fun to work out the plus-table. Subtraction is done by adding the negative, which can be computed by changing the sign of every trit. Multiplication works trit-by-trit through the multiplier, shifting at each trit.

Your task is to write functions that perform arithmetic on balanced ternary integers. 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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Pounds, Shillings, Pence

April 7, 2015

Mixed-radix number systems have a base, or radix, that varies at each position. For instance, the old-style British pounds, shillings and pence form a mixed-radix system where there are twelve pence in a shilling and twenty shillings in a pound, and calendars form a mixed-radix system where there are sixty seconds in a minute, sixty minutes in an hour, twenty-four hours in a day, and seven days in a week.

Your task is to write a program that accepts a definition of a mixed-radix system — for instance, (7 24 60 60) for the calendar mentioned above — and performs addition and subtraction of numbers written in that system. 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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