- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

Can you design a new shape for the twenty-eight squares and arrange the numbers in a logical way? What patterns do you notice?

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

Exactly 195 digits have been used to number the pages in a book. How many pages does the book have?

Challenge Level

Well here are five Aliens, all from different planets.

They all count using one "hand" except the last one who uses his six tentacles.

So let me introduce them:

1st - Serious "Dubo" who counts in twos;

2nd - The two headed "Threebee" who counts in threes;

3rd - The laughing "Quort", who counts in fours;

4th - The green-headed "Chindi" who (you guessed it) counts in fives;

Finally - The four-eyed "Senda" counting in sixes.

They've come to Earth to learn about us and what we do!

They visit a school and are interested in how many earth children there are in each room.

Dubo starts counting like this (up to what we would call seven and record as "7"):

Then the next two:

Threebee | and then Quort |

Then the last two aliens, Chindi and Senda, do their counting:

When we do our counting we have units, tens and hundreds. Those hundreds are ten lots of ten.

Notice that when Chindi counts he can only use 0, 1, 2, 3 and 4 whereas we use 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

The aliens go to a classroom and count the children. We would count 30 - but they would record and say that differently.

Here's how they record:

Dubo 11110; 1 (2 lots of 2 lots of 2 lots of 2), 1 (2 lots of 2 lots of 2), 1 (2 lots of 2), and 1 (lot of 2)

Threebee 1010; 1 (3 lots of 3 lots of 3), and 1 (lot of 3)

Quort 132; 1 (4 lots of 4), 3 ( lots of 4), and 2

Chindi 110; 1 (5 lots of 5), and 1 (lot of 5)

Senda 50; 5 (lots of 6)

It would be a good idea to try to see how their counting goes after (our) "7", what they'd say and how they'd record it.

When we humans try to look at what the aliens had written we might put a mask over most of the digits, all except the second from the right!

So when for example we try to look at one of Quort's numbers (suppose 0 0 1 3 1) a mask is over it and all we see is the 3.

Find numbers that the aliens used to count the children in the classroom that look the same

to the humans looking at the masks.

For example, to start you off:

When we look at Quort's 3 it looks the same as Senda's 35, or Chindi's 33 etc.

Find all the pairs you can.

Here's a help for either checking or to set you thinking about further explorations.

So, if we saw 1 in the unmasked window then these would be the possibilities:

Plenty to explore here!

Pupils can explore all kinds of things around these five different counting methods.

You can arrange a whole period of time for being a Threebee. For example calculating and saying and recording all numbers in that style.

These different ways of counting are also known as "bases". We use base 10, and many things connected with electrical engineering use base 2.

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.

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

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