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Broad Topics > Numbers and the Number System > Place value

Six Times Five

Stage: 3 Challenge Level:

How many six digit numbers are there which DO NOT contain a 5?

Even Up

Stage: 3 Challenge Level:

Consider all of the five digit numbers which we can form using only the digits 2, 4, 6 and 8. If these numbers are arranged in ascending order, what is the 512th number?

Lesser Digits

Stage: 3 Challenge Level:

How many positive integers less than or equal to 4000 can be written down without using the digits 7, 8 or 9?

Arrange the Digits

Stage: 3 Challenge Level:

Can you arrange the digits 1,2,3,4,5,6,7,8,9 into three 3-digit numbers such that their total is close to 1500?

Permute It

Stage: 3 Challenge Level:

Take the numbers 1, 2, 3, 4 and 5 and imagine them written down in every possible order to give 5 digit numbers. Find the sum of the resulting numbers.

Not a Polite Question

Stage: 3 Challenge Level:

When asked how old she was, the teacher replied: My age in years is not prime but odd and when reversed and added to my age you have a perfect square...

X Marks the Spot

Stage: 3 Challenge Level:

When the number x 1 x x x is multiplied by 417 this gives the answer 9 x x x 0 5 7. Find the missing digits, each of which is represented by an "x" .

Digit Sum

Stage: 3 Challenge Level:

What is the sum of all the digits in all the integers from one to one million?

Basically

Stage: 3 Challenge Level:

The number 3723(in base 10) is written as 123 in another base. What is that base?

Pupils' Recording or Pupils Recording

Stage: 1, 2 and 3

This article, written for teachers, looks at the different kinds of recordings encountered in Primary Mathematics lessons and the importance of not jumping to conclusions!

Reasoned Rounding

Stage: 1, 2 and 3 Challenge Level:

Four strategy dice games to consolidate pupils' understanding of rounding.

Mini-max

Stage: 3 Challenge Level:

Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . .

Stage: 2 and 3 Challenge Level:

Watch our videos of multiplication methods that you may not have met before. Can you make sense of them?

Back to the Planet of Vuvv

Stage: 3 Challenge Level:

There are two forms of counting on Vuvv - Zios count in base 3 and Zepts count in base 7. One day four of these creatures, two Zios and two Zepts, sat on the summit of a hill to count the legs of. . . .

Really Mr. Bond

Stage: 4 Challenge Level:

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?

Seven Up

Stage: 3 Challenge Level:

The number 27 is special because it is three times the sum of its digits 27 = 3 (2 + 7). Find some two digit numbers that are SEVEN times the sum of their digits (seven-up numbers)?

Football Sum

Stage: 3 Challenge Level:

Find the values of the nine letters in the sum: FOOT + BALL = GAME

Skeleton

Stage: 3 Challenge Level:

Amazing as it may seem the three fives remaining in the following `skeleton' are sufficient to reconstruct the entire long division sum.

Phew I'm Factored

Stage: 4 Challenge Level:

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.

Just Repeat

Stage: 3 Challenge Level:

Think of any three-digit number. Repeat the digits. The 6-digit number that you end up with is divisible by 91. Is this a coincidence?

What an Odd Fact(or)

Stage: 3 Challenge Level:

Can you show that 1^99 + 2^99 + 3^99 + 4^99 + 5^99 is divisible by 5?

Tis Unique

Stage: 3 Challenge Level:

This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only possibility.

Eleven

Stage: 3 Challenge Level:

Replace each letter with a digit to make this addition correct.

What a Joke

Stage: 4 Challenge Level:

Each letter represents a different positive digit AHHAAH / JOKE = HA What are the values of each of the letters?

Big Powers

Stage: 3 and 4 Challenge Level:

Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.

Always a Multiple?

Stage: 3 Challenge Level:

Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...

Stage: 3 Challenge Level:

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

Enriching Experience

Stage: 4 Challenge Level:

Find the five distinct digits N, R, I, C and H in the following nomogram

Nice or Nasty

Stage: 2 and 3 Challenge Level:

There are nasty versions of this dice game but we'll start with the nice ones...

Cayley

Stage: 3 Challenge Level:

The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"?

Exploring Simple Mappings

Stage: 3 Challenge Level:

Explore the relationship between simple linear functions and their graphs.

Stage: 3, 4 and 5

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.

Reach 100

Stage: 2 and 3 Challenge Level:

Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.

Legs Eleven

Stage: 3 Challenge Level:

Take any four digit number. Move the first digit to the 'back of the queue' and move the rest along. Now add your two numbers. What properties do your answers always have?

Never Prime

Stage: 4 Challenge Level:

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

Repeaters

Stage: 3 Challenge Level:

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

Latin Numbers

Stage: 4 Challenge Level:

Can you create a Latin Square from multiples of a six digit number?

Three Times Seven

Stage: 3 Challenge Level:

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

Multiplication Magic

Stage: 4 Challenge Level:

Given any 3 digit number you can use the given digits and name another number which is divisible by 37 (e.g. given 628 you say 628371 is divisible by 37 because you know that 6+3 = 2+7 = 8+1 = 9). . . .

Two and Two

Stage: 3 Challenge Level:

How many solutions can you find to this sum? Each of the different letters stands for a different number.

Cycle It

Stage: 3 Challenge Level:

Carry out cyclic permutations of nine digit numbers containing the digits from 1 to 9 (until you get back to the first number). Prove that whatever number you choose, they will add to the same total.

Number Rules - OK

Stage: 4 Challenge Level:

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

2-digit Square

Stage: 4 Challenge Level:

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

Quick Times

Stage: 3 Challenge Level:

32 x 38 = 30 x 40 + 2 x 8; 34 x 36 = 30 x 40 + 4 x 6; 56 x 54 = 50 x 60 + 6 x 4; 73 x 77 = 70 x 80 + 3 x 7 Verify and generalise if possible.

Novemberish

Stage: 4 Challenge Level:

a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.

DOTS Division

Stage: 4 Challenge Level:

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

Balance Power

Stage: 3, 4 and 5 Challenge Level:

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?

Back to Basics

Stage: 4 Challenge Level:

Find b where 3723(base 10) = 123(base b).

Plus Minus

Stage: 4 Challenge Level:

Can you explain the surprising results Jo found when she calculated the difference between square numbers?