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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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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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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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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1) The numbers generated have an equal number of 0-bits and 1-bits.

2) The maximum run of consecutive 1-bits is consistent with probability theory.

Your task is to write a simple random number tester. 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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Your task is to write a program to find prime chains. 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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Write a program that finds the maximum product of three numbers in a given array of integers.

We studied that problem in a previous exercise, where unfortunately we got it wrong. Here is the suggested solution from that exercise:

(define (max-prod-three xs) (let ((len (length xs)) (xs (sort < xs))) (cond ((< len 3) (error 'max-prod-three "insufficient input")) ((= len 3) (apply * xs)) ((positive? (car xs)) (apply * (take 3 (reverse xs)))) ((negative? (last xs)) (apply * (take 3 (reverse xs)))) ((and (negative? (car xs)) (positive? (cadr xs))) (apply * (take 3 (reverse xs)))) ((and (negative? (cadr xs)) (negative? (caddr (reverse xs)))) (* (car xs) (cadr xs) (last xs))) ((and (negative? (cadr xs)) (positive? (caddr (reverse xs)))) (max (apply * (take 3 (reverse xs))) (* (car xs) (cadr xs) (last xs)))) (else (error 'max-prod-three "missed case")))))

Your task is to write a correct program that finds the maximum product of three numbers in a given array of integers; you might start by figuring out what is wrong with the previous program. 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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int s = 0; for (int i=0; i<x; i++) for (int j=i+1; j<y; j++) for (int k=j+1; k<z; k++) s++;

What is *s*? What is a closed-form formula for computing *s*?

Your task is to compute *s*. 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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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.

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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.

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