Prize Maths Quiz: Five Circles in a Triangle (PMQ7)

Five Circles in a Triangle

To end a week of geometry problems, it is no surprise that PMQ7 is also a geometric puzzle.

The diagram shows five circles, each with integer radius, all touching the base of the large triangle. The four smaller circles all touch their two neighbouring circles, with the large circle touching all four. The two sides of the triangle each touch two of the circles.

Let the radii of the circles be a, b and c, such that a > b > c.

Given that c has a length of 4 units, find the area of the large triangle. You may leave it as an exact solution.

As always, I would like to see a method. As an interim solution, find the radii of all three circles.

Good luck!

Prize Maths Quiz: A Tricky Tower (PMQ6)

Alice was feeling sick and lying in bed in the middle of the afternoon. She hated feeling sick and she hated lying in bed. Well, that wasn’t entirely true; she loved feeling soft and warm in her bed, but today she was feeling hot and irritated.

Bill suddenly bounced through the door.

“Hello Alice! How are you? I just popped round to cheer you up a bit – Mother said you had a slight fever.”

Bill was being chirpy; it seemed like bad manners to be chirpy on a day when Alice wasn’t.

“Look, I brought you a present!” Bill handed Alice a big box covered in coloured triangles and shapes. Before Alice had a chance to ask, Bill volunteered to explain, “It’s a jigsaw puzzle. A big jigsaw puzzle! I hope you like it.”

Alice thought Bill wanted a pat on the head and a fresh bone. The pattern on the front looked interesting but, as Alice opened the box, she frowned.

Midweek Maths: You are in a maze of twisty little passages, all alike

This week I’ve created my own adventure game. I didn’t mean to, it’s just turned out that way. In the name of finding some amusing (and frustrating) strategic games I headed towards some old favourites, only to discover that they had been dismantled and sold as scrap.

In 1984 Infocom and Douglas Adams released an adventure game version of the Hitchhiker’s Guide to the Galaxy. Employing the same wit and tangential narrative of the original books and radio plays, the game was a huge hit. Then graphics arrived and people forgot how to read and write and the age of text-based interactive fiction, better known as adventure games, became a chapter in geek history.

BBC Radio 4 (yes, radio!) did have a graphical version of this game, which worked about a year ago (or maybe two) but has been terminated. Ironic that pressing the “DON’T PANIC!” button now reveals the laconic phrase “zmachine xml invalid”; this is as terminal as it gets. So, if anybody can find a working version of this game hiding in some corner of the net, then let us know, otherwise it’s back to the Douglas Adams page for the original text version.