Posts tagged Art of Problem Solving
AoPS: Candyland

Infinitely many math beasts stand in a line, all six feet apart, wearing masks, and with clean hands.

Since Grogg is generous, he decides to give away his n pieces of candy. He gives one piece of candy to each of the next n beasts in line and then leaves the line.

The other beasts repeat this process: the beast in the front, who has k pieces of candy, passes one piece each to the next k beasts in line and then leaves the line.

For some values of n, another beast (besides Grogg) temporarily holds all the candy. For which values of n does this occur?

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UKMT: Six-Digit Integers

How many six-digit integers have an even number of even digits?

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AoPS: Resigned Resolutions

Since I always fail my New Year's resolutions, this year, my New Year's resolutions are:

1. Make a new year's resolution
2. Fail my new year's resolution

Am I guaranteed to succeed? Guaranteed to fail? Something else?

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Parallel (Simon Singh): Palindromic products

What happens if we add one more 1?

What is 1111111111 × 1111111111?

Of course, you can type this into a calculator, but first try to follow the pattern and work out what the answer might be… then check it with a calculator.

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Nrich: Rhombuses

How many rhombuses made up of two adjacent small triangles are there?

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UKMT: Circle

What is the largest number of dots that the circle can pass through?

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Parallel (Simon Singh): Rotten Riddle

What am I?

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UKMT: How Many Squares?

How many squares of any size are there in this 8 x 8 grid?

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The Institute of Mathematics and its Applications (IMA): Fighting Cancer with Mathematics

Mathematics Matters... Find out how mathematicians are Fighting Cancer with Maths in the Mathematics Matters

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Nrich: Fraction of a Square

What fraction of the square is shaded?

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Parallel (Simon Singh): Partitioning

How many ways can you partition 4?

How many ways can you partition 5?

How many ways can you partition 6?

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AoPS: Weird Equations 2

Let weird(n) be the number of ways to write n as a + a + b + c, where n, a, b, and c are distinct positive integers. Is weird(n) an increasing function?

Surprisingly, weird(n) is not increasing... it sure seems like it should be! See if you can find a pattern for when weird(n) goes down. Can you explain what is going on?

Hint: the author wrote a script to print the first few values of n where weird(n) goes down and a few other things before the pattern became apparent! This problem is a good example of how computers can help us find patterns!

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AoPS: Triangles

Two copies of the same right triangle. What's the missing length?

@Cshearer41

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Nrich: Ten numbers = 37

Can you pick ten numbers from the bags that add up to 37?

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Parallel (Simon Singh): The Grand Old Duke of York

What percentage of the 10,000 men were still there when they reached the bottom of the hill?

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AoPS: Weird Equation

Let weird(n) be the number of ways to write n as a + a + b + c, where n, a, b, and c are distinct positive integers. Is weird(n) an increasing function?

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UKMT : Alice & Charlotte

In how many years time will Alice be twice as old as Charlotte will then be?

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AoPS: Polygon Symmetry

Start with a convex regular polygon P. Make a new convex polygon Q by using only some of the vertices of P.

Is it possible to make a polygon Q in this way so that Q has rotational symmetry, but no reflectional symmetry?

An example P and Q is shown here, with Q drawn in red.

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Harriet Ball: No problem today, just a video of an amazing maths teacher

I first came across Harriet Ball many years ago. One of her daughters sadly took all of Harriet’s videos down. They used Harriet’s methods to develop the curriculum for the USA KIPP Public Charter Schools network.

I use several of her times tables methods with my own students.

One very special lady, who sadly died at too young an age.

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UKMT: Counters

In how many different ways could I place the two counters?

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