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You are on one side of a river, and with you, there is a wolf, a goat and a cabbage. You have one boat, and can only take one living thing at a time. The goat cannot be left alone with the cabbage and the wolf cannot be left alone with the goat. How many journeys must you do in minimum to get all the objects to the other side of the river? In how many different ways can you do it?

Here is a very interesting way to look at the problem which involves representing the problem in 3D wolf/goat/cabbage space.

The problem is then changed to getting from one vertex of the cube to another.

Finally, we looked at another river crossing puzzle, this time involving wildebeest and lions. This puzzle is fully explained and answered in this Ted-Ed talk.

Before looking at how they propose to solve the puzzle, how did you go about solving it?

The first task of today was if numbers were chosen at random, what would be the probability that numbers started with the digits 1 to 9. Of course the probability would be the same for each digit.

Ok, let’s see if this works. Go ahead and find any article, it can be a news article, a scientific article, etc. Now count the number of times that each digit 1 to 9 is at the start of a number. Does the distribution seem even?

Let’s see if this trend can be found in mathematical sequences too. Here’s an example: the Fibonacci sequence.

1,1,2,3,5,8, …

Let’s find the probability of numbers starting with digits 1 to 9 in the first 10,000 numbers of this sequence.

Quick tip: you can use excel to do this and then use the countif function to find the distribution of probabilities of numbers that start with each digit. Does the distribution seem even?

This strange pattern of occurrence is called Benford’s law, and it is very counter-intuitive.

If you would like to learn more, click on this link: https://brilliant.org/wiki/benfords-law/

So here is your final challenge: every single time you see a number today, write it down and you can then verify the law for yourself. How close did you get?

Welcome back to Maths Club!

Today, we looked at a few logic problems based on the popular game, Mastermind. If you have never played it before, you can try online here.

We then answered these interesting questions  about what you can deduce from certain opening moves.

For the answers to these questions and some of the mind bending maths behind Mastermind, have a look at this paper by Tom Davis.

It is the end of the year and the Lycee and we would like to wish everyone a very enjoyable summer.

Please keep enjoying maths over the summer … we recommend the 100 Day Challenge on www.brilliant.org. Can you solve the first puzzle above?

We bid for chocolates via a Silent Auction. The winner of each round was the person who bid the highest unique bid. You can see how we got in the spreadsheet above!

Finding betting ways to vote

We looked at different ways to decide which candidate or choice won a vote. We realised that by using different methods, different candidates won the vote. This raises the troubling question about whether our way of voting is the best and the fairest. Here is a link to the activity we did and here is a link to an interactive website that discuss precisely this question. In the UK, the voting system is relative majority. However, it could be a good thing if we use absolute majority voting, because it would allow members of smaller parties to also get a voice. This could lead to a bigger number of parties rather than two or three major parties. However, there is no perfect voting system.

Gabriel wrote the numbers 1-9 in a 3×3 grid.

He then multiplied together all the numbers in each row and wrote the resulting product next to that row.
He also multiplied the numbers in each
column together, and wrote the product
under that column.
He then rubbed out the numbers 1-9.

Can you work out where Gabriel placed the numbers 1-9?

Did you have more information than you needed?

This puzzle comes directly from this NRich page, with the pdf of the problem available here.

Today we looked at this “playable post on the nature of society”. It is a great activity to make you think about how you can positively contribute to society becoming less segregated.

We tried to work out the maths behind the applet. Where does 95% come from in the screenshot from above?? In the javascript for the applet you can see that a measure of “sameness” is used which involves the fraction of neighbours that are the same as you. Can you create/recreate the algorithm used to find the 95%?

 

91 card game

This card game can be played with two or more players. The goal of the game is to collect the most number of points possible. You do this by biding on the diamonds, using your cards.

The game starts with each player having one complete suit, for instance, one player could have the spades. You can use more than one pack if there are more than three players.

The diamonds are mixed up and placed in the middle. These diamonds are turned over one by one and each player throws a card depending on the value of the diamond card. The values for each card are as follows:

King – 13   Queen – 12   Jack – 11 and the rest of the cards follow as such, Ace being worth 1 point.

The player who’s card has the highest value collects the diamond card. In the picture above, the person who played the King of Clubs would win the Jack of Diamonds.

If there is a tie and two or more people play cards of the same value, then another diamond is added to the middle and all the players bid again for these cards. If the final card played by each player is a tie then no one wins the diamonds in the middle.

The player who wins is the player who collects the most points out of the 91 available after all cards have been played.

Can you find a winning strategy to be able to collect the most points every time?

Another great puzzle from MathPickle … can you find the winning starting position?  What happens somewhere between 20 and 30?

Polypad can be used to display some ideas on the computer, here is a screenshot of one of our combinations: