Number Grid Explorer — Interactive Tool

1

Shade the multiples of 3. Describe the pattern you see. Now change the grid to have 9 columns. What happens to the pattern and why?

2

Shade the multiples of 2, then also shade the multiples of 5. Which numbers are shaded by both? What do you notice about them?

3

If I shade multiples of 4 and multiples of 6, will every multiple of 12 be shaded by both? Explain your reasoning.

4

Change the grid settings to start at 3 with a step of 3. What do you notice about the grid? Which multiples buttons would have no effect now, and why?

5

With a 10-column grid, multiples of 5 make two vertical columns. How many columns would multiples of 5 make on a 15-column grid? What about a 7-column grid?

6

Shade multiples of 9 on the default 10-column grid. Look at the diagonal pattern. Now try a 9-column grid. Explain what changes and why.

7

Is there a number whose multiples would shade every cell on a standard 1–100 grid? What is the largest number whose multiples shade at least 10 cells?

8

Shade the prime numbers. Apart from 2, what do you notice about whether primes are even or odd? Can you explain why?

9

Look at the primes on the default 10-column grid. Why are there no primes in the column containing 4, 14, 24, 34…? Which other columns can never contain a prime (apart from the first row)?

10

Change the grid to 6 columns and shade the primes. You should notice the primes cluster in just two columns. Which columns, and why?

11

Are there any primes between 24 and 28? Between 90 and 96? Use the grid to find the biggest gap between consecutive primes in the first 100 numbers.

12

Twin primes are primes that differ by 2 (like 11 and 13). How many pairs of twin primes can you find up to 100? Do you think twin primes ever stop?

13

Shade the square numbers. What do you notice about the gaps between consecutive squares? (1, 4, 9, 16…) Can you predict the next square number without calculating it?

14

Shade both the square numbers and the odd numbers. What do you notice? Can you find a square number that is not the sum of consecutive odd numbers?

15

Shade the triangular numbers (1, 3, 6, 10, 15…). Pick any two consecutive triangular numbers and add them together. What kind of number do you always get?

16

Which numbers are both square and triangular? Can you find all of them up to 100? Do you think there are more beyond 100?

17

Shade the cube numbers. How many perfect cubes are there between 1 and 100? How many between 1 and 1000? (Change the grid to start at 1 with step 1 and use 20 rows and 20 columns to explore.)

18

Shade both Even and Odd at the same time. What do you notice? Will this always be true regardless of the grid settings? Test it.

19

Change the start number to 2 and the step to 2. Shade Odd. What happens? Now try start 1, step 2. Explain the result.

20

On the default grid, shade Even and also shade Primes. Only one number is shaded by both. Which is it, and why is it special?

21

Shade the Fibonacci numbers. How many are there between 1 and 100? Why do they become more and more spread out?

22

Shade both Fibonacci and Odd. Are most Fibonacci numbers odd or even? Can you spot a pattern in which Fibonacci numbers are even?

23

Shade both Fibonacci and Prime. Which Fibonacci numbers are also prime? Do you think there are infinitely many Fibonacci primes?

24

Set the grid to 12 columns and shade multiples of 3 and multiples of 4. Compare the patterns to those on a 10-column grid. Why do the patterns change?

25

Create a grid starting at 0 with step 1. Shade the primes. Is 0 prime? Is 1 prime? Research why mathematicians made these decisions.

26

Set the grid to start at 1 with a step of 2 (so it shows only odd numbers). Shade the primes. How does this change your view of where primes appear?

27

Create a grid with start 1, step 1, and 6 columns. Shade the multiples of 2 and multiples of 3. Now describe a number that would remain unshaded. What kind of numbers are these?

28

Goldbach’s Conjecture says every even number greater than 2 is the sum of two primes. Pick five even numbers from the grid and try to write each as the sum of two primes. Can you find an even number where this is difficult?

29

Using manual shading, colour all the numbers from 1 to 100 that can be written as the sum of two square numbers. What proportion of numbers have this property? Are there any patterns in the numbers you can’t shade?

30

Design your own grid investigation. Choose a grid size, a starting number, and a step size that creates an interesting pattern when you shade a particular category. Write up what you discover.