'Something to think about': Mathematical problems from Mason, Burton & Stacey, Thinking Mathematically (2nd Edition)

It helps to be doing mathematical problem solving to get a sense of how it works for both learners and teachers.

As you work on these puzzles, note with your 'teacher bird' awareness how your 'learner bird' is engaging with the problem solving!

1) Thirty-one:
Two players alternately name a number from 1, 2, 3, 4 or 5. The first player to bring the combined total of all the numbers announced to 31 wins. what is the best number to announce if you go first?

- Try playing it!
-What do you need to record?
-What totals enable you to win in one move? Generalize!
-Can you find a strategy for playing that will guarantee you to win?

extend:
•What if 31 is changed to some other number?
•What if the permitted numbers are 1, 2, 3, 4, 5 and 6?
•What if there are 3 players?
•What if the permitted numbers are 1, 3, 5 or 2, 3, 7?

2) Sticky Angles:
Given a supply of sticks, all the same length, and a supply of angles all the same size, can you join the sticks together end to end at the given angle to make a closed ring?

-Have you tried it physically?
-Introduce a way of making a supply of angles!
-Have you stayed in the plane?
-Have you tried folding a strip of paper appropriately to mimic many sticks joined at the correct angle?
-Will your method always work, or is your angle special in some way?

extend:
•What is the shortest such sequence when it is possible?
•What length sequences are possible?
•Does it help to have more than one angle available, particularly when you are confined to the plane?

3) Sequence:
Write down a sequence of 0s and 1s. Underneath each consecutive pair write a 0 if they are the same and a 1 if not. Repeat this process until you are left with a single digit. Can you predict what the final digit will be?

-Specialize systematically.
-Allow your system to alter as you begin to see what is going on.
-Try to be systematic about patterns and not about lengths of sequences.
-Try working backwards from the final digit.
-Find a convincing argument to support your conjecture.

extend:
•Write down a sequence of 0s and 1s in a circle and proceed as before.
•Set your result in a more general context by using 0, 1 and 2 with some appropriate rule.

4) Milk crate
A certain square milk crate can hold 36 bottles of milk. Can you arrange 14 bottles in the crate so that each row and column has an even number of bottles?

-Depict the crate. Find a way to be able to manipulate bottle substitutes.
-Specialize to crates of other sizes
-How many bottles might there be in each row and column?
-Specializing to larger crates might help.
-Eventually you will deal with the 36-bottle crate, but what about square crates in general?
-Can you build up new arrangements from old ones?

extend:
•Try to place other numbers of bottles.
•Try rectangular crates.
•What is the largest/ smallest number of bottles that can be suitably arranged in a given crate?
•How many ways are there to place the bottles?