If a number N is expressed in binary by using only 'ones,' what can
you say about its square (in binary)?
A composite number is one that is neither prime nor 1. Show that
10201 is composite in any base.
Find b where 3723(base 10) = 123(base b).
Let N be a six digit number with distinct digits. Find the number N given that the numbers N, 2N, 3N, 4N, 5N, 6N, when written underneath each other, form a latin square (that is each row and each. . . .
Explore a number pattern which has the same symmetries in different bases.
Show that the infinite set of finite (or terminating) binary
sequences can be written as an ordered list whereas the infinite
set of all infinite binary sequences cannot.
A collection of games on the NIM theme
Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.
Using balancing scales what is the least number of weights needed
to weigh all integer masses from 1 to 1000? Placing some of the
weights in the same pan as the object how many are needed?
This article for the young and old talks about the origins of our number system and the important role zero has to play in it.
Start with any number of counters in any number of piles. 2 players
take it in turns to remove any number of counters from a single
pile. The winner is the player to take the last counter.
Have you seen this way of doing multiplication ?
We are used to writing numbers in base ten, using 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Eg. 75 means 7 tens and five units. This article explains how numbers can be written in any number base.
Explore the factors of the numbers which are written as 10101 in
different number bases. Prove that the numbers 10201, 11011 and
10101 are composite in any base.
How can Agent X transmit data on a faulty line and be sure that her message will get through?
In 'Secret Transmissions', Agent X could send four-digit codes error free. Can you devise an error-correcting system for codes with more than four digits?
Find all 3 digit numbers such that by adding the first digit, the
square of the second and the cube of the third you get the original
number, for example 1 + 3^2 + 5^3 = 135.
You have worked out a secret code with a friend. Every letter in the alphabet can be represented by a binary value.
Ask a friend to choose a number between 1 and 63. By identifying which of the six cards contains the number they are thinking of it is easy to tell them what the number is.
An example of a simple Public Key code, called the Knapsack Code is
described in this article, alongside some information on its
origins. A knowledge of modular arithmetic is useful.