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?

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?

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!

AoPS: Triangles

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

@Cshearer41

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?

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.

AoPS: Rectangle Shaded

What fraction of this rectangle is shaded?

@Cshearer41

AoPS: A Flip Side

Every card in this deck has a number on one side and a letter on the other. The same number can appear on more than one card.

Euna places four of these cards in a row, then flips over some (maybe all) of the cards and mixes them up. The before and after state is pictured below. What number is on the other side of the A?

AoPS: Jewellery Design

10 silver beads and 10 gold beads are arranged randomly on a necklace.

Is it always possible to make one straight line cut that divides the necklace into two pieces that each have 5 silver and 5 gold beads?

Year 8 Maths Homework: Find The Radius

Could you solve this? My year 8 daughter’s homework from last weekend. I did point her in the right direction and then she finished it off. She was very happy as she worked out how to solve it.

Last night, she asked if she could work on really hard maths over half term, to help her find maths at school easy and also to have fun (apparently). I’ve pulled out UKMT’s Intermediate Problems by Andrew Jobbings (years 9-11 Intermediate Maths Challenge questions) and a pile of Art of Problem Solving books.

AoPS: Sum 25 Product

A set of positive integers has sum 25. What is the biggest you can make the product of the numbers?

AoPS: The Four Cards

Shawn has dealt four cards in front of you. He claims that if a card has an even number on one side of it, then the other side of the card is blue. Which cards do you need to turn over, in order to confirm if Shawn is telling you the truth?

Art of Problem Solving (AoPS): True or False

Each statement is either true or false. How many of the following statements are true?

1. The answers to statements 2 and 3 are different.

2. The answers to statements 3 and 4 are different.

3. The answers to statements 4 and 5 are different.

4. The answers to statements 5 and 6 are different.

5. The answers to statements 6 and 7 are different.

6. The answers to statements 7 and 8 are different.

7. The answers to statements 8 and 1 are the same.

8. The answers to statements 1 and 2 are the same.

AoPS: Don't Flip Out

Alex has 25 discs that are black on one side and white on the other. He arranges them on a 5 x 5 grid so that all the white sides are showing.

A move consists of taking any three consecutive discs in a row or column and flipping them over. You want the discs to make the checkerboard coloring shown below.

From the initial all-white position, what is the smallest number of moves needed to get to the checkerboard position?

Art of Problem Solving: Triangles in Triangles

A circle is inscribed in an equilateral triangle and an equilateral triangle is inscribed in that circle. What is the ratio of the areas of the two triangles?

LANDO EDucation

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