All three puzzles are based on the same principle, similar
to the 15-puzzle or Rubik's cube. There is a display which
shows a configuration of numbers. There are several moves in each
puzzle (see below on more about the moves). Try out the moves
first. Then click on the ``Randomize'' button and try to use
the moves to restore the puzzle to its original position
(which can, should you get lost, always be obtained by clicking on 
the ``Reset'' button). 

Moves: In M12, the Invert button (I) reverses the order of numbers.
The Merge button (M) is a ``card shuffle'' best understood by trying
it out.

In M24, there are no buttons, but clicking anywhere on the black
area and dragging your mouse turns the numbers in the circle (this
move is called R or L, depending on whether you  drag to the right or
left).  Clicking on the yellow button in the middle switches every
pair of numbers with the same color (S).

In Dotto, there are four moves. This puzzle includes the M24 puzzle.
Look at the yellow/blue row in the bottom. This is, in fact, M24,
but the numbers are arranged in a row instead of a circle. The
R move is the ``circle rotation to the right'': the column above
the number 0 stays put, but the column above the number 1 moves
to the column over the number 2 etc. up to the column over the 
number 23, which moves to the column over the number 1. You may
also click on a column number and then on another column number
in the bottom row, and the ``circle rotation'' moving the first
column to the second occurs. The M move is the switch, in each
group of 4 columns separated by vertical lines (called tetrads)
the ``yellow'' columns switch and the ``blue'' columns switch.
The sign change move (S) changes signs of the first 8 columns
(first two tetrads). The tetrad move (T) is the most complicated:
Subtract in each row from each tetrad 1/2 times the sum of the numbers in
that tetrad. Then in addition to that, reverse the signs of
the columns in the first tetrad. 



Advanced notes: each of the three puzzles allows ``custom moves''.
Those are any sequences of the moves described above, which you
may store in memory, to avoid having to type them 
over and over again. The first hint to solving the puzzles
is, in fact, to design suitable custom moves. You may have
an unlimited number of custom moves. To create a custom move,
click on the ``Create Custom Move'' window, then in CAPITAL
LETTERS type the sequence of letters representing the moves
you wish to perform (the leftmost move is performed first).
Additionally, you may follow the letter by a number which means
the move will be performed the specified number of times.
(For example, in M24, R13 turns the circle to the right 13 times.)
Your custom move is stored in memory once executed at least once
by clicking on the Custom button. It can then be performed any
time by clicking on the Create custom move window, and scrolling
down to the custom move you wish to use. 

Note also the move log window: to save time typing custom moves,
you may click on the log window, type ctrl-c and then use ctrl-v
to paste the sequence to a custom move! That way you may use
existing custom moves to make new long ones very fast.



Strategy hints: Although these puzzles are similar in design
to the ``15-puzzle'' or Rubik's cube, the same strategy as
used for those puzzles (creating custom moves which only
change around very few numbers) won't work. Instead, almost
all moves will change around almost everything!

Here are strategy hints suitable for these puzzles:

M12: this game has 12.11.10.9.8 configurations, and any 5 numbers
can be (in any order) moved to any 5 places. Therefore, if you
restore the positions of the numbers 1,2,3,4,5 (or any 5 numbers),
you are done.

M24: this game has 24.23.22.21.20.48 configurations (notice the
number 48 in the end). It is still true that any 5 numbers may
be moved to any 5 places, but after you fix the position of
5 numbers, there are still 48 possible positions left.

Dotto: Notice that the sum of squares of the numbers in each row
doesn't change. (This sum of squares is 64 in the first row, 32 in every other
row.) If you manage to get an ``8'' in the first row, you have
almost reduced the game to M24 except those signs. To have the
original position, signs of all numbers on the diagonal must be +.
Hint on signs: if the only thing wrong are signs on the diagonal,
and only 8 signs are wrong, those 8 columns can be moved to the
first 8 columns by using only the M24 moves (M,R).


If you manage to solve these puzzles, you have learned, without
having to do any advanced algebra, the nuts and bolts of three
SPORADIC SIMPLE GROUPS: The Matthieu groups M12 and M24, and
the Conway group .0 (which is actualy two times the sporadic group .1).
These puzzles are representations of those groups.