Skip over navigation
Cambridge University Faculty of Mathematics NRich logo
menu search
  • Teachers expand_more
    • Early years
    • Primary
    • Secondary
    • Post-16
    • Events
    • Professional development
  • Students expand_more
    • Primary
    • Secondary
    • Post-16
  • Parents expand_more
    • Early Years
    • Primary
    • Secondary
    • Post-16
  • Problem-Solving Schools
  • About NRICH expand_more
    • About us
    • Impact stories
    • Support us
    • Our funders
    • Contact us
  • search

Or search by topic

Number and algebra

  • The Number System and Place Value
  • Calculations and Numerical Methods
  • Fractions, Decimals, Percentages, Ratio and Proportion
  • Properties of Numbers
  • Patterns, Sequences and Structure
  • Algebraic expressions, equations and formulae
  • Coordinates, Functions and Graphs

Geometry and measure

  • Angles, Polygons, and Geometrical Proof
  • 3D Geometry, Shape and Space
  • Measuring and calculating with units
  • Transformations and constructions
  • Pythagoras and Trigonometry
  • Vectors and Matrices

Probability and statistics

  • Handling, Processing and Representing Data
  • Probability

Working mathematically

  • Thinking mathematically
  • Developing positive attitudes
  • Cross-curricular contexts

Advanced mathematics

  • Decision Mathematics and Combinatorics
  • Advanced Probability and Statistics
  • Mechanics
  • Calculus

For younger learners

  • Early Years Foundation Stage

Transposition Cipher

Age 11 to 16
Challenge Level Yellow star
  • Problem
  • Getting Started
  • Student Solutions
  • Teachers' Resources

A transposition cipher is one which rearranges the order of the letters in the ciphertext (encoded text), according to some predetermined method, without making any substitutions.

Suppose we want to encrypt the following message, a quote from 'Peter Rabbit' by Beatrix Potter: "Now run along and don't get into mischief, I'm going out." Let's remove the punctuation and the spaces between the words to get: "nowrunalonganddontgetintomischiefimgoingout".

This is 44 letters long. For reasons we'll soon discover, let's add 4 extra padding characters, "x",  at the end to now get: "nowrunalonganddontgetintomischiefimgoingoutxxxx".

We can now write this message in 4 rows, each 12 letters long.

n o w r u n a l o n g a
n d d o n t g e t i n t
o m i s c h i e f i a m
g o i n g o u t x x x x

By reading the letters in order down the columns, instead of along the rows, we get:

"nnog odmo wdii rosh uncg ntho agiu leet otfx niix gnax atmx"

We can now send this message to our friend with the spaces removed, and the message is "hidden".

Suppose the enemy intercepts and wants to decipher our message. What might they do?



Once you have thought about how to decipher a message encoded in this way, read on below:

 

48 characters can be encoded using grids of one of these dimensions:

$1\times48, 2\times24, 3\times16, 4\times 12, 6 \times 8, 8 \times 6, 12 \times 4...$

The first of these doesn't rearrange the message at all.
The second size gives: 

n n o g o d m o w d i i r o s n u n c g n t h o
a g i u l e e t o t f x n i I x g n a x a t m x

Reading down the columns gives "nangoigu....". Definitely not English! 

The next arrangement is a 3 by 16 grid:

n n o g o d m o w d i i r o s n
u n c g n t h o a g i u l e e t
o t f x n i I x g n a x a t m x

"nuonntocf", let's try again!

A 4 by 12 grid gives:

n n o g o d m o w d i i
r o s n u n c g n t h o
a g i u l e e t o t f x      "nrannogi..."
n i I x g n a x a t m x

and a 6 by 8 grid gives: 

n n o g o d m o
w d i i r o s n
u n c g n t h o      "nwuaog.."
a g i u l e e t
o t f x n i I x
g n a x a t m x

Hmm, let's keep trying! An 8 by 6 arrangement gives:

n n o g o d
m o w d i i
r o s n u n   "nmrcaena...."
c g n t h o
a g i u l e
e t o t f x
n i I x g n
a x a t m x

A 12 by 4 sized grid gives:

n n o g
o d m o
w d i i
r o s n
u n c g
n t h o 
a g i u   "nowrunalongand....", the start of our original message! 
l e e t   
o t f x
n i I x
g n a x
a t m x

 

Can you see why we chose a 48-character message rather than a 44-character message?

 

Imagine you have intercepted the message below, and you know it has been encrypted using a transposition cipher.
Can you decrypt the message?
 

ttanopnshonstpdeendoaiherltsmnemaihuogrebkedmhsdbendeeetiadenrlottin
tsfbhupltefeonpyolaalnettflveedhhblewlsaieirefutnfnynodakogdtrdlarde
sseibeoetoncoswprmleuhnwaeyhteweiwdasfhlgaodtoalhywnoutx

You might find it useful to work on squared paper.  

If you want to work on a computer to solve the problem, you can download the message as a text file which doesn't contain any line breaks.

There is a transposition solver as part of our Cipher Challenge Toolkit.

 

If you are interested in code breaking you might enjoy the Secondary Cipher Challenge.

 

Notes and Background

Many codebreakers use frequency analysis as their first 'tool'. If the distribution of letters in the cipher text does not reflect the usual distribution (with E, T, A and so on as the most common letters) it is likely that a substitution or more complex encryption has been used. If the distribution reflects what you would expect for standard text, it could be that the text has simply been transposed, as it has in this problem.

Of course, with very short cipher texts, it is difficult to get any meaningful data from a frequency analysis.
For more on frequency analysis and substitution ciphers, see this problem.

You may also like

The Best Card Trick?

Time for a little mathemagic! Choose any five cards from a pack and show four of them to your partner. How can they work out the fifth?

Substitution Cipher

Find the frequency distribution for ordinary English, and use it to help you crack the code.

Secret Transmissions

How can Agent X transmit data on a faulty line and be sure that her message will get through?

  • Tech help
  • Accessibility Statement
  • Sign up to our newsletter
  • Twitter X logo

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.

NRICH is part of the family of activities in the Millennium Mathematics Project.

University of Cambridge logo NRICH logo