Question 11: Everyone is Equal

In 2011, this M4 mathematics competition had only 10 competitors. The points awarded are the same as on the current Rules page. After 5 games everyone had the same number of points!

Is this possible? If so, provide one possible solution with the maximum possible total points. (Yes, everyone on zero points is a solution, but not maximal!)

The solution here must be in a grid showing the 10 competitiors and their point score for each of the 5 questions. There are many solutions, you need to find just one.

The easiest way is to use a spreadsheet so you can automatically calculate the totals.

Spreadsheets are really useful in questions such as this. If you don't know how to use one, and especially how to enter formulas into the cells, then come back in August when the full website will start. If you can't wait, then try here or here.

Good Luck!




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Question 12: The Crimson Room

To end our short journey through mathematical puzzles, here is a classic logic game: The Crimson Room.
 
The original can be found here:

http://crimson-room.net/

The website functions but the link to the actual online game does not seem to work as of writing. However, The Crimson Room is now available as a mobile app.

Enjoy!

... and remember, solving something yourself gives you a greater buzz than searching for someone else's answer!!

One day, you will need to solve something that nobody has solved.

Question 5: A Dark Bridge

Here is another online puzzle. This time, unlike Question 4, we can write the solution in a symbolic form. The aim is to get all the people across the bridge before the lantern burns out. Instructions are on the first page.

Go to the bridge puzzle.

To write your answer, let's look at a nice and simple way to represent the people crossing the bridge. We shall represent each person by the number of minutes it takes them to cross the bridge.

If 3 and 8 cross together let's write this as (3,8).
If 6 crosses the bridge alone we can represent this with just one number (6).
So, for example, if 3 and 8 cross the bridge first, then 8 walks back alone, then 1 and 12 cross the bridge, then 3 walks back alone, we can show this simply as:
(3,8)(8)(1,12)(3)
This is obviously not a solution, and the puzzle has not been finished, it is just to show you a simple way for you to write the final solution.

A nice thing about this representation is that we can calculate the total minutes by adding the largest number in each bracket. In my example above, the total would be: 8 + 8 + 12 + 3 = 31. Way too high!

So, your final solution will be a string of brackets showing the sequence of your moves plus the toal number of minutes.

Good luck!

Question 4: Mouse Trap

Mathematics is about far more than numbers and shapes - it is fundamentally about patterns, and such patterns come in many forms. Imagine looking at a game of chess as it evolves from move to move. Every move changes the pattern on the chess board. At the end of the game the rules of chess determine whether the final pattern is a victory for white, for black or ends in a draw.

The mathematics of game theory aims to analyse games so as to find winning strategies. It has a wide range of applications in fields such as economics, biology and psychology. Now, chess is a game with simple rules yet complex strategies; far too hard for us to start with! So let's look at something simpler.

One of the favourite games at my after-school club is Mouse Trap. It is a simple game where a computer-operated mouse tries to escape from a hexagonal grid while you try to encircle it by blocking its path with bricks. You win if you manage to trap the mouse so it can no longer move; you lose if the mouse manages to reach the edge of the grid and thus escape.

Go to Mouse Trap now!


Play the game a few times and get a feel for how to trap the mouse.

This question is to write out a winning strategy. Be as precise as possible so that someone could write a computer program that takes your rules and applies them to every specific situation. This does not have to be a long essay! You should be able to describe your winning strategy in just four or five sentences.

One additional question: does your strategy win all the time?

Have fun!

Question 1: The Smallest of the Lowest

This is really a warm-up question as we wait for all the guests to arrive.

A) Take the number 180 and list the pairs of factors. Which pair of factors of 180 has the smallest Lowest Common Multiple? Your answer must include the pair of numbers and their LCM.

B) Then, which pairs of factors of 6300 has the smallest Lowest Common Denominator? Your answer must include the pair of numbers and their LCM.

C) 10 Bonus Points for the first person only to submit a short and clear explanation of the mathematical principle that can be used to answer questions such as (A) and (B) without listing all the factor pairs.

You are only allowed to submit your answers ONCE. I will take your first answers and ignore the rest. It is therefore part of the game for you to decide whether you wish to just answer questions A and B or spend a little longer to gain the Bonus Points by answering A, B and C. Let me be clear, you cannot answer questions A and B first and later submit your answer to C. Of course, you could, but I will ignore it!

Good Luck!



Remember, your answers must be submitted below in the comments box. Your answers will not be visible until the question is Closed and you are then free to discuss the solution.

First Question Coming Up!

The first question in our online maths competition is coming up on Thursday at around midday.

Just a reminder to please sign up as a Friend on the right-hand column so that you can answer the questions. As this blog is hosted by Google, you can always answer questions if you have a Google account.

It is also a good idea to sign up for email notification so you know when the next question is online plus any announcements and to discuss the answers.

My own fault for this being a little slow, so will leave questions open until we have at least 10 correct answers... from 10 different people!!

See you tomorrow!

Richard