How Mathematicians Do It – mathsquad

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How Mathematicians Do It

These problems and solutions have been created in collaboration with problem solving guru Professor Derek Holton

Introduction

A research mathematician has many tools at their disposal and uses them to explore problems and obtain new results.

The following questions are designed to expose students to a number of problem-solving techniques, and beyond that create experiences that closely resemble those of a research mathematician.

The problems have been scaffolded to maximise opportunities for success. Each problem is also accompanied by an extension question (see “Further Explorations”) to highlight the fact that mathematicians, and students, can always keep pushing problems further.

Further information on the problem solving techniques and suggestions for how to use these sets can be found here.

Full solutions are provided at the top of each question set and you can share these with students once they have had a red hot go!

The Introductory Problems
We begin with the following 5 problems due to their accessibility and the joy we have seen students get from seeing a more playful side of mathematics. We hope these problems ignite a love of problem-solving with your students too!

Solutions to this question set can be found here.

Problem 1

Leo the rabbit is climbing up a flight of steps. Leo can only hop up one or two stairs each time he hops. He never hops down, only up.

How many different ways can Leo hop up a flight of two steps?
How many different ways can Leo hop up a flight of three steps?
How many different ways can Leo hop up a flight of four steps?
How many different ways can Leo hop up a flight of five steps?
How many different ways can Leo hop up a flight of six steps?
How many different ways can Leo hop up the flight of ten steps?

Further exploration: What if Leo could hop one, two or three steps at a time?

Problem 2

Imagine a long thin strip of paper stretched out in front of you, left to right. Imagine folding this piece of paper so that the left end meets the right. Now press the strip flat so that it is folded in half with a crease. Repeat the whole operation on the new strip two more times.

How many creases are there after one fold?
How many creases are there after two folds?
How many creases are there after three folds?
How many creases are there after ten folds?
How many creases are there after n folds?

Further exploration: How many creases would there be if you alternated between horizontal and vertical folds?

Problem 3

1. There are three lily pads and a blue frog and a red frog.We want the frogs to swap sides. There can only be one frog on a lily pad at a time. Frogs can jump onto an empty lily pad that is next to them or jump over a frog to an empty lily pad. How many jumps does it take to make the swap? [You may like to create frogs and lily pads from scrap paper]

2. There are five lily pads and two blue frogs and two red frogs.

We want them to swap sides. There can only be one frog on a lily pad at a time. Frogs can jump onto an empty lily pad that is next to them or jump over a frog to an empty lily pad. How many jumps does it take to make the swap? What is the smallest number of jumps this could take?

3. What if there were three frogs of each colour? What is the smallest number of jumps this could take now?

4. What if there were four frogs of each colour? What is the smallest number of jumps this could take now?

5. Make a prediction for what happens when there are five frogs of each colour. Check your prediction by acting it out.

Further exploration: What if there were n frogs of each colour? What is the smallest number of jumps this could take now?

Problem 4

Consider the 3×3 grid shown on the right.
1. How many 1×1 squares are on this grid?
2. How many 2×2 squares are on this grid?
3. How many 3×3 squares are on this grid?
Consider a 4×4 grid.
1. How many 1×1 squares are on this grid?
2. How many 2×2 squares are on this grid?
3. How many 3×3 squares are on this grid?
4. How many 4×4 squares are on this grid?
It was once claimed that there are 204 squares on an ordinary chessboard. Can you justify this claim?

Further exploration: How many rectangles are on a chessboard?

Problem 5

There are four discs on the bottom of this triangle. How many discs are there altogether?
How many discs would there be altogether if there were ten discs on the bottom?
How many discs would there be altogether if there were 100 discs on the bottom?
How many discs would there be altogether if there were n discs on the bottom?

Further exploration: Consider triangles with rows containing only even numbers of discs. What happens now? What if instead only odd numbers of discs were used? What other patterns could you explore?

Some More Problems
These problems continue to utilise the problem-solving strategies introduced at first. They aren’t as playful, though they are very rewarding and satisfying to solve!

Solutions to this question set can be found here.

Problem 6

Using only 5¢ and 10¢ coins, in how many ways can you make 20¢?
Using only 5¢ and 10¢ coins, in how many ways can you make 50¢?
Using only 5¢ and 10¢ coins, in how many ways can you make 100¢?
Using only 5¢ and 10¢ coins, in how many ways can you make 100n ¢?

Further exploration: What happens when you have 5c, 10c and 20c coins?

Problem 7

The pages in a book are numbered starting with page 1. One digit is used to represent the numbers from 1 to 9, whereas two digits are used to represent the numbers from 10-99 (and so on).

If there are 10 pages in a book. How many digits are used?
If there are 100 pages in a book. How many digits are used?
If there are 183 pages in a book. How many digits are used?

Further Exploration: If there are n pages in a book where n=1000a+100b+10c+d. How many digits are used?

Problem 8

How many 2 digit numbers have six as the sum of their digits?
How many 3 digit numbers have six as the sum of their digits?
How many 4 digit numbers have six as the sum of their digits?

Further Exploration: How many n-digit numbers have a digit sum of k (where k<10)? Show Answers

Problem 9

What is the sum of the underlined digits, that is the ones digits, of 10, 11, 12,…, 19?
What is the sum of the ones digits of 20, 21, 22,…, 29?
What is the sum of the ones digits of all 2 digit numbers, 10, 11, 12,…, 99?
What is the sum of all the tens digits of the 2 digit numbers?
Now, what is the sum of the digits of all 2 digit numbers, 10 to 99?
What is the sum of the digits of all 3 digit numbers, 100 to 999?

Further Exploration: Wendy tried to find the formula for the sum of the digits of all n digit numbers, this is what she got: 45 x 10ⁿ + 9 x n x 10^n-1. Is she right? Either convince us that she is correct, or fix the formula and convince us why it works.

Problem 10

Below shows the different ways we can make a line out 5c and/or 10c coins that totals 20c

Lay out the 5c and 10c coins that make up 30c in a line. How many orders are there for these coins?
Lay out the 5c and 10c coins that make up 40c in a line. How many orders are there for these coins?
Lay out the 5c and 10c coins that make up 600c in a line. How many orders are there for these coins?

Further exploration: Consider what happens when we have 5c, 10c and 20c coins.

Problem 11

In the following questions, we consider a 4×1 rectangle and a 1×4 rectangle as the same rectangle. Whereas a 2×2 rectangle and a 1×4 rectangle are different rectangles (yes, a square is a rectangle!). The some approach is taken for rectangular prisms and so a 1x1x4, a 1x4x1 and a 1x1x4 are all the same rectangular prism too.

How many different rectangles of area 25cm² can be made using 1cm² tiles? List them all.
How many different rectangles of area 64cm² can be made using 1cm² tiles? List them all.
How many different rectangular prisms of volume 20cm³ can you make using 1cm³ blocks? List them all.
How many different rectangular prisms of volume 60cm³ can you make using 1cm³ blocks? List them all.

Further exploration: A plong is made up of blocks in the first dimension, blocks in the second dimension, blocks in the third dimension and blocks in the fourth. The size of a plong is calculated by multiplying the dimensions: a x b x c x d. How many different plongs of size 60 exist? What about even higher dimensions? Of course, plongs, just like rectangles and rectangular prisms are considered the same if the dimensions are the same but just in a different order.

Problem 12

What is the final digit of 2⁵?
What is the final digit of 2²⁰?
What is the final digit of 2²⁰²³?
What is the final digit of 3²⁰²³?

Further Exploration: Do the powers of other numbers have interesting patterns?

Problem 13

There are some cubes on the table. Alice and Blair alternatively remove one or two cubes. The winner is the person who takes the last cube. On the principle of ‘ladies first’, Alice always takes the first turn. Alice and Blaire are expert players, and depending on the number of cubes they will both know the winner before the even game begins.

Who wins when there is one cube?
Who wins when there are two cubes?
Who wins when there are three cubes?
Who wins when there are four cubes?
Who wins when there are 21 cubes?
Who wins when there are 31 cubes?
How do Alice and Blaire know who will win?

Further Exploration: Blocks are placed arbitrarily in two piles. When it’s their turn Alice and Blair can take either one or two blocks provided they all come from one pile. The winner is the person with the last block. Again Alice goes first. Under what circumstances will Alice win? What if there were more piles?

Problem 14

How many factors does 2 have?
How many factors does 4 have?
How many factors does 8 have?
How many factors will 2ⁿ have?
How many factors will 3ⁿ have?
How many factors will 4ⁿ have?
How many factors will 10ⁿ have?

Further exploration: The table below shows the product of the numbers in the grey cells, and then in brackets shows how many factors the number has. Complete the table below and use it to explore the number of factors of 2ⁿx3^m.

What about 2ⁿx3^mx5^r? etc..

Enabling, Extending and Generalising Problems on your own
The above problems have given you a taste of some effective strategies mathematicians use:

Acting it Out
Working Systematically
Start with simple cases and finding a pattern
Extend and generalise

Can you use these skills to answer and extend these problems?

Solutions to this question set can be found here.

Problem 15

The following are the requirements for a stack of cans to be neat:

The stack of cans has at least two rows
The stack of cans is piled so that each new row has one fewer can than the row below it
What numbers can be neatly stacked?

Further Exploration: Why can some number be neatly stacked but others can’t?

Problem 16

At a farm there are chooks and sheep. If there are 24 animals and 80 legs, how many sheep are there?
How could you extend this problem?

Problem 17

How many matchsticks are required to make the 3 by 4 grid shown below?
How could you extend or generalize this problem?

Problem 18

When 2/7 is written as a decimal, what number is in the 30th decimal place?
How could you extend or generalise this problem?

Problem 19

A cube made of 27 red cubes is dipped in yellow paint so every face is covered.

Of the 27 cubes, how many will have zero yellow faces? How many will have one yellow face? How many will have two yellow faces? How many will have three yellow faces? How many will have four yellow faces?
How could you extend or generalise this problem?

Problem 20

A fraction with a numerator of 1 is called a unit fraction. Here are different sums of unit fractions which make 1/6:

How many ways can you write 1/8 as the sum of two different unit fractions?
How could you extend or generalise this problem?

The Final Set of Problems

Each of these new problems are connected in some way to one of the problems above. We wonder if you can see which problems each of these connects to? You might even spot connections we didn’t – how cool would that be!!

Solutions to this question set can be found here.

Problem 21

If there are 4 people in a room and everyone was to fist bump every other person, 6 fist bumps would occur. How many fist bumps would occur if there were 100 people in the room?
Note: A fist bump is a “young person’s handshake” 🙂

Problem 22

There are spiders, pigs and chooks on a farm. There are 32 legs, how many of each animal could there be?

Problem 23

We have to build a footpath, 60cm by 6m, out of bricks that are 30cm by 60cm. The bricks can lie vertically and horizontally, but in no other direct. Three possible brick arrangements are shown below. How many different ways are there to build the walkway?

Problem 24

To solve the puzzle of the Tower of Hanoi, you are required to move all disks to Tower 3 whilst never placing a larger disk onto a smaller disk.

a. What is the smallest number of moves in which this puzzle can be completed?

b. What if there were 10 disks instead of 3? What is the smallest number of moves in which this puzzle can be completed?