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Broad Topics > Sequences, Functions and Graphs > Iteration

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Sums and Products of Digits and SP Numbers

Stage: 5

This article explores the search for SP numbers, finding the few that exist and the proof that there are no more.

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Try to Win

Stage: 5

Solve this famous unsolved problem and win a prize. Take a positive integer N. If even, divide by 2; if odd, multiply by 3 and add 1. Iterate. Prove that the sequence always goes to 4,2,1,4,2,1...

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Spirostars

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

A spiropath is a sequence of connected line segments end to end taking different directions. The same spiropath is iterated. When does it cycle and when does it go on indefinitely?

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Peaches in General

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

It's like 'Peaches Today, Peaches Tomorrow' but interestingly generalized.

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V-P Cycles

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Form a sequence of vectors by multiplying each vector (using vector products) by a constant vector to get the next one in the seuence(like a GP). What happens?

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Climbing Powers

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

$2\wedge 3\wedge 4$ could be $(2^3)^4$ or $2^{(3^4)}$. Does it make any difference? For both definitions, which is bigger: $r\wedge r\wedge r\wedge r\dots$ where the powers of $r$ go on for ever, or. . . .

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Stretching Fractions

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Imagine a strip with a mark somewhere along it. Fold it in the middle so that the bottom reaches back to the top. Stetch it out to match the original length. Now where's the mark?

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Weekly Challenge 48: Quorum-sensing

Stage: 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

This problem explores the biology behind Rudolph's glowing red nose.

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Rain or Shine

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Predict future weather using the probability that tomorrow is wet given today is wet and the probability that tomorrow is wet given that today is dry.

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Infinite Continued Fractions

Stage: 5

In this article we are going to look at infinite continued fractions - continued fractions that do not terminate.

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Difference Dynamics

Stage: 4 and 5 Challenge Level: Challenge Level:1

Take three whole numbers. The differences between them give you three new numbers. Find the differences between the new numbers and keep repeating this. What happens?

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Difference Dynamics Discussion

Stage: 5

This article discusses what happens, and why, if you generate chains of sequences getting the next sequence from the differences between the adjacent terms in the sequence before it, eg (7, 2, 8, 3). . . .

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Slippage

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

A ladder 3m long rests against a wall with one end a short distance from its base. Between the wall and the base of a ladder is a garden storage box 1m tall and 1m high. What is the maximum distance. . . .

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The Golden Ratio, Fibonacci Numbers and Continued Fractions.

Stage: 4

An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.

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Recent Developments on S.P. Numbers

Stage: 5

Take a number, add its digits then multiply the digits together, then multiply these two results. If you get the same number it is an SP number.

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Triangle Incircle Iteration

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Start with any triangle T1 and its inscribed circle. Draw the triangle T2 which has its vertices at the points of contact between the triangle T1 and its incircle. Now keep repeating this. . . .

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Route to Root

Stage: 5 Challenge Level: Challenge Level:1

A sequence of numbers x1, x2, x3, ... starts with x1 = 2, and, if you know any term xn, you can find the next term xn+1 using the formula: xn+1 = (xn + 3/xn)/2 . Calculate the first six terms of this. . . .

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First Forward Into Logo 10: Count up - Count Down

Stage: 3, 4 and 5 Challenge Level: Challenge Level:1

What happens when a procedure calls itself?

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Dalmatians

Stage: 4 and 5 Challenge Level: Challenge Level:1

Investigate the sequences obtained by starting with any positive 2 digit number (10a+b) and repeatedly using the rule 10a+b maps to 10b-a to get the next number in the sequence.

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Stringing it Out

Stage: 4 Challenge Level: Challenge Level:1

Explore the transformations and comment on what you find.

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First Forward Into Logo 1: Square Five

Stage: 2, 3 and 4 Challenge Level: Challenge Level:1

A Short introduction to using Logo. This is the first in a twelve part series.