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There are **23** NRICH Mathematical resources connected to **Iteration**, you may find related items under Patterns, sequences and structure.

Problem
Primary curriculum
Secondary curriculum
### Climbing Powers

$2\wedge 3\wedge 4$ 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 $(r^r)^r$, where $r$ is $\sqrt{2}$?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Rain or Shine

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### A Very Shiny Nose?

This problem explores the biology behind Rudolph's glowing red nose, and introduces the real life phenomena of bacterial quorum sensing.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Archimedes Numerical Roots

How did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

Age 16 to 18

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Difference Dynamics Discussion

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) maps to (5, 6, 5, 4).

Age 16 to 18

Problem
Primary curriculum
Secondary curriculum
### Difference Dynamics

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?

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Ford Circles

Can you find the link between these beautiful circle patterns and Farey Sequences?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Peaches in General

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Spirostars

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?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Stretching Fractions

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?

Age 14 to 16

Challenge Level

Article
Primary curriculum
Secondary curriculum
### The Golden Ratio, Fibonacci Numbers and Continued Fractions.

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.

Age 14 to 16

Problem
Primary curriculum
Secondary curriculum
### Slippage

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 up the wall which the ladder can reach?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### V-P Cycles

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?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Dalmatians

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.

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Stringing it Out

Explore the transformations and comment on what you find.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### First Forward Into Logo 10: Count up - Count Down

What happens when a procedure calls itself?

Age 11 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### First Forward Into Logo 1: Square Five

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

Age 7 to 16

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Infinite Continued Fractions

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

Age 16 to 18

Problem
Primary curriculum
Secondary curriculum
### Converging Means

Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the two sequences.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Differs

Choose any 4 whole numbers and take the difference between consecutive numbers, ending with the difference between the first and the last numbers. What happens when you repeat this process over and over again?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Loopy

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Triangle Incircle Iteration

Keep constructing triangles in the incircle of the previous triangle. What happens?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Route to Root

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 sequence. What do you notice? Calculate a few more terms and find the squares of the terms. Can you prove that the special property you notice about this sequence will apply to all the later terms of the sequence? Write down a formula to give an approximation to the cube root of a number and test it for the cube root of 3 and the cube root of 8. How many terms of the sequence do you have to take before you get the cube root of 8 correct to as many decimal places as your calculator will give? What happens when you try this method for fourth roots or fifth roots etc.?

Age 16 to 18

Challenge Level