Mosaic Help
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How to solve Mosaic Puzzles Mosaic puzzles (formerly known as fill-a-pix) are rapidly growing in popularity around the world. They offer the opportunity for some solvers to develop new logical skills and for others to learn new techniques. This brief introduction is in response to requests for help from readers of Tsunami and Enigma who find the puzzles a bit daunting at first sight. First of all, of course, go for the obvious fills and empties – 0 or 9 anywhere in the grid, 4 in a corner and 6 along an edge are all straightforward.
0 x x x
4 x x x x 0 x x x x 9
(Diagram 1) Filling in X’s by a nought effectively creates a little ‘edge’ so a 6 next door to that ‘edge’ can be filled in This leads us into a method for using two numbers together: 0 and 6, two cells apart is obvious. Slightly less so is 1 and 7. The fill for the 1 has to be in the three cells next to the 7 otherwise, we couldn’t complete the 7 at all. So we can mark in six empty cells where the 1 cannot be and fill in six of the 7. We cannot yet work out where that seventh fill will go but that will become clear as the neighbouring cells are completed. A moment’s thought will show that this same logic applies to 2 and 8 and (trivially) 3 and 9. All these pairs of numbers differ by six.
(Diagram 2) Looking at pairs of numbers that are adjacent to each other we find another pattern. A 3 next to a 6 gives three fills and three empties. Of the six cells by the side of the 3 and below, only three can be filled. So the 6 must use the three cells below that number. As the three fills are in the six cells just mentioned, the three cells above the 3 must be empty.
(Diagram 3) So 4 next to 7 will behave in a similar fashion. As will 5 and 8 and (again, trivially, 6 and 9). Note that these pairs of numbers all differ by three. This sort of thinking can then be applied to other pairs of numbers when there is an empty cell or a fill already marked. We will leave you to work out what can be filled or marked empty for the 5/3 pair in the bottom-left corner of diagram 3. As you solve the puzzle and apply this kind of logic that uses two numbers together, you will find other arrangements that will let you make certain fills and empties.
6 x x x x 0 x x x x 6
x x x x 1 x 7
8 9 x 3 x x x x
x x x 3 6
x x x 4 7
x x x 5 8
x x x 6 9 3 5
x x 2 x x x
A G u i d e To M o s a i c P u z z l e s ( I I ) In Part One last month we found that pairs of numbers could be used together to make definite fills and mark in definite empties (X). What was even more useful was that these pairs also formed an easily remembered pattern: 1/4 2/5 3/6 4/7 5/8 (6/9).
X
7 2
Only two of the cells inside the lined square can be filled and as the 7 only has two empty cells altogether, they must be within that square. So the fills for the 2 are known to lie within that square and all its other neighbours can be marked empty. We can also fill in all the other cells around the 7, since we know its only two empty cells are within the marked area.
3 5
456 789
X
7
X
2
X
X
X
Last issue’s leftovers: This edge position was left as a question in Mosaic Issue One with an invitation to work out the logical fills and empties.
To find further pairs that will behave in exactly the same way, we can add or subtract from the pair we have found. This gives us: 1/6 2/7 3/8 Any of these pairs of numbers diagonally adjacent to each other will produce the same filling and marking of empty cells. We can commit this to memory as the Diagonal 1/6 Pattern. The 5 must use three of the four cells shared by it and the 3. So we can place X’s around the rest of the 3’s cells and fill the two to the left of the 5.
X
123 456
The 6/9 pattern is trivial really since filling in the nine automatically completes the six, so we can ignore cases where one number is nine or zero. We can memorize this simply as the 1/4 Pattern and extend it in our heads by successively adding one to each of the pair. The result is the same:
So when we spot another pair that work together, it is worth pausing to see if they too are part of a larger logical picture. For example:
X
X
5
X
X
3
X X
And we can logically extend this pattern to include, by subtracting one from each number, 2/4 and 1/3. So now we can add the Edge Diagonal 1/3 to our armoury of logical patterns.
A G u i d e To M o s a i c P u z z l e s ( I I I ) In the last two issues we found that pairs of numbers could be used together to make definite fills and mark in definite empties (X). What combinations of numbers are useful also depends on whether we are working in a corner, along an edge or in the centre of the grid. There is one corner pattern (3/1) still to be mentioned:
X 3
1
3
X
1X X
Only one of the cells in the second column can be filled. So the corner cell and the one above must be filled and those around the 1 must be empty.
The same logic can be applied to give us another edge pattern – the Edge 5/1. The fill for the 1 must be between it and the 5. So we can mark in four X’s and four fills. Previously, we have extended patterns by adding to each number in the pair. Here though, we can add to the power of this pattern by realising that the 1 can migrate around the 5 provided we keep two cells in between the numbered cells.
X 5
1
5
1X X X
X
X
1X X
1 5
1
X
5
X
X
X
X
And
5 X
X
X
X 1X X X 5
To end this session, it is worth noting that edge patterns can arise in the centre of a grid because we have an artificial wall of X’s to one side of a number. Here is just such a situation: The X’s given by the 0 effectively put the 5 along an edge, so the Edge 5/1 pattern still works.
X
X 1X X
1
X 5 X
X
5 X
X 0X X
X
X
X
X 0X X
A G u i d e To M o s a i c P u z z l e s ( I V ) In the last three issues we found that pairs of numbers could be used together to make definite fills and mark in definite empties (X). What combinations of numbers are useful also depends on whether we are working in a corner, along an edge or in the centre of the grid. There are a couple more centre patterns that are worth remembering, in particular because the two numbers are not directly adjacent and are easier to overlook. First, the 1/7 pattern:
Of the three squares in the row between the 1 and the 7, only one can be filled. So we know that two of these are empty and so the six squares above must be filled for the 7. And six x’s can be entered around the 1. As before, when a pattern works for one pair of numbers, it will also work for a similar pair – so 2/8 will produce the same logical result.
7
7
1
X 1X X X
8
2
X 2X X X
X
A further logical situation emerges when there is a 1 a knight’s move away from the 8:
8
X
X
8
8
X 1
1X X X
It’s also useful to remember that a fill next to a number effectively reduces its value by one and may then bring it within the scope of a known logical pattern.
X
Or:
8
8
2
X 2X X
The fill by the 2 reduces it to a 1. So the situation is the same as above.
8
8
1
X X1 X
X
A G u i d e To M o s a i c P u z z l e s ( V ) Before we move onto using three numbers at once, there is a fairly obvious two-number edge pattern that needs recording. We call it the Edge1/1. The fill for the 1 along the edge must be in the six shaded squares and these all take care of the other 1 in the second row. So the three cells below that 1 must be empty. It’s not much, but it’s something and those X’s may be vital to the area below.
As before, when a pattern works for one pair of numbers, it will also work for a similar pair – so 2/2, 3/3, 4/4 and 5/5 will all produce the same logical result: three X’s below the second number. 6/6 is the trivial upper limit to this pattern. In issues one and two we met the edge pattern where 5 and 3 were diagonally adjacent. There is also a useful situation where they are orthogonally adjacent along an edge. Only one of the four shaded cells next to both the 1 and the 3 can be filled, so the other two cells for the 3 must be on its right and there are two empty cells to the left of the 1. Once again, if this patterns works then so does 2/4 and 3/5 adjacent along an edge.
X
9 7
9
X
1
1
1
1
1
3
X
X
X
1
3
X
X
A useful way into finding patterns is to start from a trivial situation and work backwards. For a first three-number combination, here is the filling when a 7 is sandwiched between two 9’s.
If the cell containing the 5 were empty, then the lightly shaded squares would have to be filled but that would give a block of six, not five. So that cell must be filled. Of the three shaded cells between it and an 8 exactly two are filled so the other two cells next to the five must be empty and the others around the 8’s must be filled. By subtraction 7/3/7 and 6/1/6 laid out in the same way will produce the same pattern of fills and crosses.
X
8 5
8
X
0 x x x
4 x x x x 0 x x x x 9
(Diagram 1) Filling in X’s by a nought effectively creates a little ‘edge’ so a 6 next door to that ‘edge’ can be filled in This leads us into a method for using two numbers together: 0 and 6, two cells apart is obvious. Slightly less so is 1 and 7. The fill for the 1 has to be in the three cells next to the 7 otherwise, we couldn’t complete the 7 at all. So we can mark in six empty cells where the 1 cannot be and fill in six of the 7. We cannot yet work out where that seventh fill will go but that will become clear as the neighbouring cells are completed. A moment’s thought will show that this same logic applies to 2 and 8 and (trivially) 3 and 9. All these pairs of numbers differ by six.
(Diagram 2) Looking at pairs of numbers that are adjacent to each other we find another pattern. A 3 next to a 6 gives three fills and three empties. Of the six cells by the side of the 3 and below, only three can be filled. So the 6 must use the three cells below that number. As the three fills are in the six cells just mentioned, the three cells above the 3 must be empty.
(Diagram 3) So 4 next to 7 will behave in a similar fashion. As will 5 and 8 and (again, trivially, 6 and 9). Note that these pairs of numbers all differ by three. This sort of thinking can then be applied to other pairs of numbers when there is an empty cell or a fill already marked. We will leave you to work out what can be filled or marked empty for the 5/3 pair in the bottom-left corner of diagram 3. As you solve the puzzle and apply this kind of logic that uses two numbers together, you will find other arrangements that will let you make certain fills and empties.
6 x x x x 0 x x x x 6
x x x x 1 x 7
8 9 x 3 x x x x
x x x 3 6
x x x 4 7
x x x 5 8
x x x 6 9 3 5
x x 2 x x x
A G u i d e To M o s a i c P u z z l e s ( I I ) In Part One last month we found that pairs of numbers could be used together to make definite fills and mark in definite empties (X). What was even more useful was that these pairs also formed an easily remembered pattern: 1/4 2/5 3/6 4/7 5/8 (6/9).
X
7 2
Only two of the cells inside the lined square can be filled and as the 7 only has two empty cells altogether, they must be within that square. So the fills for the 2 are known to lie within that square and all its other neighbours can be marked empty. We can also fill in all the other cells around the 7, since we know its only two empty cells are within the marked area.
3 5
456 789
X
7
X
2
X
X
X
Last issue’s leftovers: This edge position was left as a question in Mosaic Issue One with an invitation to work out the logical fills and empties.
To find further pairs that will behave in exactly the same way, we can add or subtract from the pair we have found. This gives us: 1/6 2/7 3/8 Any of these pairs of numbers diagonally adjacent to each other will produce the same filling and marking of empty cells. We can commit this to memory as the Diagonal 1/6 Pattern. The 5 must use three of the four cells shared by it and the 3. So we can place X’s around the rest of the 3’s cells and fill the two to the left of the 5.
X
123 456
The 6/9 pattern is trivial really since filling in the nine automatically completes the six, so we can ignore cases where one number is nine or zero. We can memorize this simply as the 1/4 Pattern and extend it in our heads by successively adding one to each of the pair. The result is the same:
So when we spot another pair that work together, it is worth pausing to see if they too are part of a larger logical picture. For example:
X
X
5
X
X
3
X X
And we can logically extend this pattern to include, by subtracting one from each number, 2/4 and 1/3. So now we can add the Edge Diagonal 1/3 to our armoury of logical patterns.
A G u i d e To M o s a i c P u z z l e s ( I I I ) In the last two issues we found that pairs of numbers could be used together to make definite fills and mark in definite empties (X). What combinations of numbers are useful also depends on whether we are working in a corner, along an edge or in the centre of the grid. There is one corner pattern (3/1) still to be mentioned:
X 3
1
3
X
1X X
Only one of the cells in the second column can be filled. So the corner cell and the one above must be filled and those around the 1 must be empty.
The same logic can be applied to give us another edge pattern – the Edge 5/1. The fill for the 1 must be between it and the 5. So we can mark in four X’s and four fills. Previously, we have extended patterns by adding to each number in the pair. Here though, we can add to the power of this pattern by realising that the 1 can migrate around the 5 provided we keep two cells in between the numbered cells.
X 5
1
5
1X X X
X
X
1X X
1 5
1
X
5
X
X
X
X
And
5 X
X
X
X 1X X X 5
To end this session, it is worth noting that edge patterns can arise in the centre of a grid because we have an artificial wall of X’s to one side of a number. Here is just such a situation: The X’s given by the 0 effectively put the 5 along an edge, so the Edge 5/1 pattern still works.
X
X 1X X
1
X 5 X
X
5 X
X 0X X
X
X
X
X 0X X
A G u i d e To M o s a i c P u z z l e s ( I V ) In the last three issues we found that pairs of numbers could be used together to make definite fills and mark in definite empties (X). What combinations of numbers are useful also depends on whether we are working in a corner, along an edge or in the centre of the grid. There are a couple more centre patterns that are worth remembering, in particular because the two numbers are not directly adjacent and are easier to overlook. First, the 1/7 pattern:
Of the three squares in the row between the 1 and the 7, only one can be filled. So we know that two of these are empty and so the six squares above must be filled for the 7. And six x’s can be entered around the 1. As before, when a pattern works for one pair of numbers, it will also work for a similar pair – so 2/8 will produce the same logical result.
7
7
1
X 1X X X
8
2
X 2X X X
X
A further logical situation emerges when there is a 1 a knight’s move away from the 8:
8
X
X
8
8
X 1
1X X X
It’s also useful to remember that a fill next to a number effectively reduces its value by one and may then bring it within the scope of a known logical pattern.
X
Or:
8
8
2
X 2X X
The fill by the 2 reduces it to a 1. So the situation is the same as above.
8
8
1
X X1 X
X
A G u i d e To M o s a i c P u z z l e s ( V ) Before we move onto using three numbers at once, there is a fairly obvious two-number edge pattern that needs recording. We call it the Edge1/1. The fill for the 1 along the edge must be in the six shaded squares and these all take care of the other 1 in the second row. So the three cells below that 1 must be empty. It’s not much, but it’s something and those X’s may be vital to the area below.
As before, when a pattern works for one pair of numbers, it will also work for a similar pair – so 2/2, 3/3, 4/4 and 5/5 will all produce the same logical result: three X’s below the second number. 6/6 is the trivial upper limit to this pattern. In issues one and two we met the edge pattern where 5 and 3 were diagonally adjacent. There is also a useful situation where they are orthogonally adjacent along an edge. Only one of the four shaded cells next to both the 1 and the 3 can be filled, so the other two cells for the 3 must be on its right and there are two empty cells to the left of the 1. Once again, if this patterns works then so does 2/4 and 3/5 adjacent along an edge.
X
9 7
9
X
1
1
1
1
1
3
X
X
X
1
3
X
X
A useful way into finding patterns is to start from a trivial situation and work backwards. For a first three-number combination, here is the filling when a 7 is sandwiched between two 9’s.
If the cell containing the 5 were empty, then the lightly shaded squares would have to be filled but that would give a block of six, not five. So that cell must be filled. Of the three shaded cells between it and an 8 exactly two are filled so the other two cells next to the five must be empty and the others around the 8’s must be filled. By subtraction 7/3/7 and 6/1/6 laid out in the same way will produce the same pattern of fills and crosses.
X
8 5
8
X
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