The notes below come from a presentation at the 2014 *Biarri Applied Mathematics Conference* in Melbourne.

#### Getting Started with the Curios

The Curios are a collection of images, each containing a hidden message. The vast majority of them use some kind of encoding of the alphabet, rather than encrypting the message, so the main task is to figure out how the message is encoded... and then decode it!

For example, the following image is actually not one of the Curios but gives a good introduction to the idea:

It doesn't take long to notice that, of the many Rudolfs, some have a black nose while others have a red nose. This suggests that the message is being encoded as a binary number. Note that 2^{5} = 32 so five binary digits, *bits*, are required to represent a letter of the alphabet (since 2^{4} = 16 is less than 26). With ten Rudolfs on each row, that suggests that each row gives two letters of the alphabet.

It then just needs to be decided as to whether black is 0 and red is 1, or red is 0 and black is 1. For the former the first two letters would be

`01000` = 8 (**H**) and `00001` = 1 (**A**)

while for the latter we would have

`10111` = 23 (**W**) and `11110` = 30 (undefined)

Since the second option gives a value beyond 26, this suggests the former is the correct encoding. You can then read off the rest of the message.

#### Curio #53 (Wisdom – Part 2)

Continuing on the binary theme, consider Curio #53:

There are elements with two possible states that appear in groups of five, suggesting a similar binary encoding to above. As before, you need to figure out which state is 0 and which is 1, and in this case whether the bits are read top to bottom or bottom to top. Trying these options suggests that the first symbol is an **A** and then again the rest of the message can be read off.

#### Curio #56 (Applied Regression)

In other cases, the binary encoding might be a little more subtle. For example, consider Curio #56:

What are the two binary states here? The points are evenly spaced in the horizontal direction but then in the vertical direction they are either above or below the line. Taking "above" as 1 and "below" as 0, the first five points give `00100` = 4 (**D**), the second five points give `00101` = 5 (**E**), the third five give `10011` = 19 (**S**), and so on.

#### Curio #65 (So Close)

Is this a binary Curio?

This is a little more challenging because you first need to recognise the original text that has been distorted by the binary code. Once you do, each letter is either "correct" (0) or "incorrect" (1). For example, the real text begins *Tis but thy name* so the first bits are `01001` = 9 (**I**), followed by `00011` = 6 (**C**), and so on.

#### Curio #67 (Special Power)

Of course not all Curios use a binary encoding (though several that are currently unsolved do). Another obvious encoding is to use a *ternary* system, such as in Curio #67:

These require figuring out how a 0, 1 and 2 are represented. (Here they are three rotations of the water symbol.)

Base 4 is not so useful for making Curios (since 4^{2} = 16 misses a lot of the alphabet while 4^{3} = 64 has too much redunancy). Base 5 features in several Curios, such as Curio #55, since 5^{2} = 25 (enough for everything except **Z**).

#### Final Notes

Not all Curios use base 2, 3 or 5 encodings. Many try some more novel ways of representing the message, such as Curio #63 (using multiples of the alphabet numbers) or Curio #75 (using elements of a permutation group for the encoding).