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This Game-Puzzle branch of the TRINPsite tree of files offers sliding
(block) puzzles, that is, two-dimensional spatial puzzles in which
you are asked to rearrange blocks on a board without lifting them.
These puzzles are interactive in that the user can order changes in them,
in this case the movement of a block from one place on the board to
another place on the board, provided that the order complies with the
rules of the game.
The interactive sliding block puzzles presently available here are:
- Uncover the Three by Three, a sliding puzzle on a
five-by-five board, with three variants:
- simplest, 6-block variant
a trial version and prototype
- 6-block unsolvable variant
just for the real puzzler who wants to be challenged (or does
not believe me)
- 7-block variant
solvable, but presumably in no fewer than 26 moves
- Uncover the Six by Six, a sliding puzzle on a
ten-by-ten board, with two variants:
- simplest, 8-block variant
a trial version
- 14-block variant
i can solve
this one in 138 moves, but you may do better
- Uncover the Five by Five, a sliding puzzle on a
thirteen-by-thirteen board, with one variant:
- simplest, 13-block variant
a trial version
These sliding block puzzles belong to just one category.
I have dubbed them "three-square cover-uncover sliding puzzles"
(or 'triscus puzzles' for short).
In these puzzles a central or near-central square is entirely covered by
another square in the initial arrangement and entirely uncovered in the
final arrangement.
(In order to prevent confusion, the little elementary squares of which the
entire board is composed will be called "cells" here.)
Upon completion the central or near-central square will show a (part of a)
text or a picture which is somehow related to TRINPsite as a whole or to
one of its branches.
The subdivision into types of three-square cover-uncover sliding puzzles
depends on the 'natural' solutions to the following mathematical equation,
in which x, y and z must be positive natural numbers:
x2 + y2 = z2
This formula is the same as the one for the Pythagorean theorem, but
instead of using it to express the relationships between the properties of
one-dimensional objects (the lengths of the three sides of a right-angled
triangle), i use it to express the relationships between the properties of
two-dimensional objects (the surfaces of three squares of which one
contains the two others).
Every 'triscus puzzle' has three defining squares in the
initial arrangement:
- the board square (with sides z)
- a covered square (with sides x) consisting of cells on the
board with a text or picture
- a covering square (with sides y) consisting of blocks to be
moved away from the square of cells to be uncovered
In the final arrangement the square with sides x is completely
uncovered, while all (!) other cells on the board are covered.
This means that, at the end, the covering square has disappeared as a
square.
Visually speaking, the covering square could be of the same size as the
covered square (x=y), but for mathematical reasons this is not possible as
we will have to confine ourselves to positive natural numbers.
Therefore, a number of cells which was covered in the initial arrangement
will still be covered in the final arrangement (but, perhaps, by a
different block).
Visually speaking, it is also much nicer if both the covering square and
the square covered by it in the initial arrangement are located exactly in
the horizontal center and the vertical middle, but this is an exception
rather than the rule.
For z>1 and z<21 the quadratic equation underlying this category of
sliding puzzles has three series of natural-number solutions: the series
starting with (3,4,5), (5,12,13) and (8,15,17) for (x,y,z).
If (a,b,c) is a solution, then (2a,2b,2c), (3a,3b,3c), etc. are solutions
too, so (6,8,10), (9,12,15) and (12,16,20) belong to the same series as
(3,4,5).
Mathematicians will recognize these solutions as Pythagorean triples.
It is these Pythagorean triples which determine the size of the whole
board (z) and of the central or near-central covered (x) and covering
squares (y) on it.
The number of moves needed to change the initial into the final arrangement
depends, of course, on the number and shape of the blocks in the covering
square, but the minimum number can be calculated for the simplest design.
(Moves are either in a vertical or in a horizontal direction,
one place at a time.
A change of place in a diagonal direction will count as two moves.)
The table below shows the first six types of three-square
cover-uncover sliding puzzles, based on the size of the board
used:
| Z | X | Y |
moves |
centered square(s) | example |
| 5 | 3 | 4 | min. 14 |
only covered sq. |
simplest var. |
| 10 | 6 | 8 | min. 24 |
both squares |
simplest var. |
| 13 | 5 | 12 | min. 57 |
only covered sq. |
simplest var. |
| 15 | 9 | 12 | ? |
only covered sq. | |
| 17 | 8 | 15 | ? |
only covering sq. | |
| 20 | 12 | 16 | ? |
both squares | |
The number of moves needed to solve this kind of sliding puzzle increases
considerably with the size of the board.
No fewer than 57 moves (and 13 blocks) are needed to solve the puzzle on a
13-by-13 board for the simplest and, perhaps, most boring layout on a board
of such a size.
Even the puzzle on the 5-by-5 board cannot be solved in fewer than 14
moves, whether designed with 6 or with 8 blocks in the covering square.
A more interesting and challenging division into blocks only makes the
minimum number of moves increase, for example from 24 to 138 on a
ten-by-ten board.
Nonetheless, these 138 moves can be made in about a quarter of an hour, so
that time need not be a reason to pull out.
Enjoy!
Vincent van Mechelen
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