Find all the different quadrilaterals you can make by joining four dots on a 9-dot grid.
Different in this case means not congruent – i.e. none of your quadrilaterals should be able to be formed by rotating, translating or reflecting one of your other quadrilaterals.
Here is a couple to get you started:
How many can you find?
Some of these puzzles are taken from a nice book by Alex Bellos called Can you solve my problems? Alex Bellos does a fun puzzle blog on the Guardian every two weeks.
The Four Stacks
Start with eight coins in a row, and create four stacks of two coins in four moves. A move consists of moving one coin to the left or right by hopping over two coins and landing on the third one along. You can hop over single coins or stacks.
Tait’s Teaser
The aim of the puzzle is to start with two different types of coin (or heads and tails of the same coin) in the arrangement above and get to the arrangement below in as few ‘moves’ as possible.
A move consists of moving two adjacent coins at the same time. You can move them anywhere in the same line, but you can’t switch the two coins around as you do so.
The answer is not five!
Frogs and Toads
Place six coins as shown above. The white ones represent toads and the grey ones are frogs. Frogs and toads can only move by hopping over one other frog or toad to an empty space. Toads can move to the right, and frogs to the left. Can you rearrange them into the position below?
See here for an interactive version of this puzzle created by NRich which allows you to change the number of frogs and toads.
Star Puzzle
Draw a star and try to place 9 coins on any of the black vertices. You can place coins by starting from a vertex which is empty, moving in a straight line and counting 1,2,3. Number 1 is the vertex you start on, number 2 may or may not have a coin on it, and number 3 is where you place your coin.
Two possible opening moves are shown below
Here is an attempt that has gone wrong! 7 coins have been placed but there is no way to place any more, because you must start on an empty vertex.
Here is a great little applet coded by Etienne Royer-Gray to play with:
The first challenge today was to cut a square out of a piece of paper using just one straight cut with a pair of scissors.
The picture above gives you a hint how to fold it first if you look closely!
Next challenge is to cut out an equilateral triangle by doing some careful folding and just one straight cut.
Here are instructions for making the letters of your name just using one cut! Click on “Folding Steps” for step by step instructions.
Paper folding and cutting can be taken very far – you can try to make all the number diamond playing cards in this way!!
For hints on this challenge and more activities see this page from the Wild Maths website.
Happy New Year! Hope you enjoyed your extra second before hand! Try this game online aptly called Approaching Midnight.
Then try Got It and Escape Vortex.
You may able to use similar strategies for all three!
This week we’ve started on a series of six James Bond themed Christmassy maths puzzles. All taken from this amazing website. Puzzles include Licence to Fill, Safe Cracker and Code Breaker.
On a special farm chickens live in circular fields, separated by fence posts and straight pieces of fencing
Is there a pattern between the number of fence posts and the maximum number of chickens that can be kept?
Are you sure? How do you know?
Can you explain why?
Hint: The answer when you have 6 posts is not what you might think!
When you have had a play around with this problem you might like to read this article or check out this solution.
Exploring the Koch Snowflake (it’s getting cold after all!)
Draw the biggest equilateral triangle you can on a page
On each line/side, cut it in three parts and remove the middle part
Add two sides of a triangle in the gap.
Repeat like so:
What develops is a fractal. If you carry on this pattern for ever the area is bounded but the perimeter is infinite! You can learn how to make the Koch snowflake on Python here.
This site gives the code for a Seirpinski Triangle. You can try it on Trinket here.
Numberphile have an amazing collection of maths videos, one relating to fractals is here. And here is a great TED talk by Benoit Mandelbrot.
Logo is a language that was written in 1967, but is still interesting to work with nearly 50 years later!
There is an online version to use or Windows users can download a free copy of FMSlogo. A good free version for Mac users is ACSlogo.
THe basic LOGO commands are:
FD forward BK backward
LT left RT right
PU pen up PD pen down
HT hide turtle ST show turtle
CS clear screen CT clear text
The following challenges are adapted from the brilliant monthly puzzle website NRich, try the full set of puzzles after.
Challenge 1
Try drawing these shape on LOGO
Challenge 2
Draw regular polygons just by using the repeat command.
e.g. REPEAT 5 [FD 30 RT 72]
Challenge 3
Now trying using nested repeats to create shapes like this!
Today we looked at strategies for solving a tricky version of a popular puzzle. Usual Sudoku rules apply – numbers 1 to 9 in every row, column and box – but in this version you don’t get any starting numbers, just the total sum of the numbers in each dotted box.
Hint: The numbers 1-9 add up to 45, see if you can use that fact and the bottom left square to calculate this number (see how one of the dotted boxes overlaps the squares):
12 + 7 + 16 = 35. This helps a lot!
Hat puzzles provide endless entertainment! The following ones which accompany the picture above are taken directly this week’s Monday puzzle in the guardian by Alex Bellos , which is an excellent source of maths puzzles.
Alice, Bob and Charlie are well-known expert logicians; they always tell the truth. They are sat in a row, as illustrated above. In each of the scenarios below, their father puts a red or blue hat on each of their heads. Alice can see Bob’s and Charlie’s hats, but not her own; Bob can see only Charlie’s hat; Charlie can see none of the hats. All three of them are aware of this arrangement.
(i) Their father puts a hat on each of their heads and says: “Each of your hats is either red or blue. At least one of you has a red hat.” Alice then says “I know the colour of my hat.” What colour is each person’s hat?
(ii) Their father puts a new hat on each of their heads and again says: “Each of your hats is either red or blue. At least one of you has a red hat.” Alice then says “I don’t know the colour of my hat.” Bob then says “I don’t know the colour of my hat.” What colour is Charlie’s hat?
(iii) Their father puts a new hat on each of their heads and says: “Each of your hats is either red or blue. At least one of you has a red hat, and at least one of you has a blue hat.” Alice says “I know the colour of my hat.” Bob then says “Mine is red.” What colour is each person’s hat?
(iv) Their father puts a new hat on each of their heads and says: “Each of your hats is either red or blue. At least one of you has a red hat, and at least one of you has a blue hat.” Alice then says “I don’t know the colour of my hat.” Bob then says “My hat is red”. What colour is Charlie’s hat?
(v) Their father puts a new hat on each of their heads and says: “Each of your hats is either red or blue. Two of you who are seated adjacently both have red hats.” Alice then says “I don’t know the colour of my hat.” What colour is Charlie’s hat?
Answers can be found here
Another great puzzle is:
Four prisoners are arrested for a crime, but the jail is full and the jailer has nowhere to put them. He eventually comes up with the solution of giving them a puzzle so if they succeed they can go free but if they fail they are executed.
The jailer seats three of the men into a line. The fourth man is put behind a screen (or in a separate room). He gives all four men party hats. The jailer explains that there are two black hats and two white hats, that each prisoner is wearing one of the hats, and that each of the prisoners see only the hats in front of him but neither on himself nor behind him. The fourth man behind the screen can’t see or be seen by any other prisoner. No communication among the prisoners is allowed.
If any prisoner can figure out what color hat he has on his own head with 100% certainty (without guessing) and tell the jailer, all four prisoners go free. If any prisoner suggests an incorrect answer, all four prisoners are executed. The puzzle is to find how the prisoners can escape, regardless of how the jailer distributes the hats.
Answer is here.
And finally the super tricky one:
100 prisoners are lined up by an executioner, who places a red or blue hat upon each of their heads.
The prisoners can see the hats of the people lined up in from of them, but they cannot look at the hats behind them, or at their own.
Starting at the back of the line, the executioner asks the last prisoner to state the colour of his hat.
In order to live, the prisoner must answer correctly.
The night before the line-up, the prisoners can discuss a strategy to help them survive. How many prisoners can be guaranteed to be saved?
See here for a discussion of the solution.