Archive for November, 2001

Romantic mathematical puzzles

Tuesday, November 6th, 2001

Those who know me will tell you that I am an incurable romantic, and also that I have a soft spot for mathematical puzzles and games. Here are some mathematical odds and ends that I have found to be unusually romantic - see what you think. Follow the links on the titles for further information.


The Happy End Problem

The Happy End Problem is so named because two of the mathematicians who met whilst working on the problem, E. Klein and G. Szekeres, got married and lived happily ever.

Suppose you have g>=3 points on the plane, no three of which are collinear.

For some arrangements of the g points, you will be able to pick n of them to make a convex n-sided polygon. For example, for n=3 you will always be able to make a convex triangle because all triangles are convex and no 3 points are collinear.

For n=4, g=4 does not suffice because the four points could be arranged as the corners of a triangle and a fourth point inside the triangle - no convex quadrilateral can be obtained from these points, but g=5 does (see figure 1).

The problem is to determine the number of points g(n) you need to make an n-sided polygon no matter where the points are (as long no 3 of them are collinear).

The first few values of g(n) are:
g(3)=3
g(4)=5
g(5)=9
g(6)=37
g(7)=128
g(8)=464
g(9)=1718


Mrs Miniver's Problem

According to Mrs Miniver, in the ideal romance each lover shares exactly two thirds of their interests with the other. She wishes to symbolize this with a diagram of two circles of the same size, overlapping such that the area of the overlap is that same as sum of the areas of each of the two crescents formed (half of the area of the overall figure). What is the ratio of the distance between the centres of the circles and their radii?

The answer is approximately 0.529864, and is believed to be trancendental.

(An earlier version of the problem stated here was to find the answer if the intersection area was the same as the area of one of the crescents, which gives the answer 0.807946, but this isn't what Mrs Miniver originally stated as the ideal romance).


Happy numbers

Pick a positive integer n. Take the squares of the digits and add them up. Repeat. If you eventually get to 1, n is happy. If you don't, n is unhappy. The first few happy numbers in base 10 are 1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100.


Friendly numbers

Two postive integers a and b are friendly if the sum of the proper factors of a is b and the sum of the proper factors of b is a.


Happy friendly numbers

Here's a romantic puzzle: in which bases (if any) can you find a pair of numbers which are friendly with each other and happy together?


The Kissing problem

An interesting article about kissing spheres.

Fractal gallery

Friday, November 2nd, 2001

For those of you who can't get enough of fractal pictures, here are some of my favorites. The orbit types were created with programs of my own devising, written with DJGPP and Allegro. The others were created with Fractint, at 1600x1200 and resampled to 800x600.

Ultima maps

Friday, November 2nd, 2001

Warning! You will find major spoilers here if you haven't completed these games yet!

Maps from Ultima VI

Britannian surface

Dungeon level 1

Dungeon level 2

Dungeon level 3

Dungeon level 4

Gargoyle surface


Maps from Savage Empire

Eodon Valley surface

Myrmidex caves

Surface level caves

Underground city of Kotl


Maps from Martian Dreams

Mars surface

Mines

Dream World

Dream World

Mine and underground parts of Martian city

Coal mine and power plant


About the maps

The scale of these maps is 1 pixel per step. The data is stored in the files MAP and CHUNKS and isn't too difficult to decode, but colouring is fiddly. To colour these maps I modified the MAP and CHUNKS files to find out what tile each number corresponds to. The tiles are as follows:

Ultima VI (left), Savage Empire (centre), Martian Dreams (right)

Note that these aren't all the graphics from the games because there are two types of tile. The other tiles are object tiles, the positions and types of which are stored in the files in the SAVEGAME folder. Going to a finer scale would be rather pointless without deciphering this data. For example, you can see even at this scale that the buildings are empty.

It would be really cool to make a 1:1 scale map including all the objects, equivalent to the Eagle Eye spell. Unfortunately it would be a bit too large for your web browser - stored as a GIF file each large map would take up at least 70Mb! However, it would look nice scaled down (even better than these maps at the same scale) and it would be good to have a program that would let you scroll around and zoom into/out of the map. It would also be nice to create a census of all the characters in the games and find out the purpose (if any) of the message "Winona Ryder is a really hot babe" in Savage Empire and Martian Dreams. I may do these things someday.

Surprising stuff the maps revealed

I was surprised that the maps in Ultima VI actually form a three dimensional structure - if you scale up the small maps to the size of the large one then all the cave entrances, holes and ladders line up. The same is not true of Savage Empire and Martian Dreams. In fact, in Savage Empire the "level" isn't even related to the 3 dimensional configuration of the world.

The authors embedded hidden information into the maps as you can see - SMB (Stephen Beeman) in Savage Empire and Gryphon (Philip Brogden) in Martian Dreams. There are also some hidden areas in the poles of Mars. I don't know if these have any purpose in the game or not - I'll tell you when I've completed it.

Although the "shape of the world" is supposed to be flat in Ultima VI and Savage Empire and spherical in Martian Dreams, the true shape of each world is a torus or doughnut. If you could walk off the left you would reappear at the same place on the right, and if you could walk off the top you would reappear at the same place on the bottom. In fact you can see this effect in the Savage Empire maps: there is a river which disappears off the top and reappears on the bottom on the surface, and the rightmost walls of the underground city are on the left. This is used in Martian Dreams to create a cylindrical (okay, spherical...) world rather than a flat one.

Savage Empire uses only 4 of the 6 levels. Level 5 is empty but level 6 contains the gargoyle world straight out of Ultima VI! It doesn't quite make sense because some of the chunks (8x8 blocks of tiles) are different, but it's clearly the same map.

Links

These maps were inspired by those of Otmar Lendl. Check them out, he's got some of Ultima IV and Ultima V as well.

Mobygames entries for Ultima VI, Savage Empire and Martian Dreams.

Bartholomew Melnicki has gone much further and started plotting objects on the maps. Check it out!