Abacus techniques, easiest to hardest.
In Dayzle the hint and the solver are one engine, so these are the 12 steps it takes to solve a Abacus board. Open one for a worked example and a clip of the hint teaching it on a real board.
Beginner
Naked Single Beginner An empty cell where every digit but one is already in its row or column, or a cell with one note left.
Go through the digits for a cell and cross off each one already in its row or its column. Say on a 4×4, Row 2 holds 1 and 4 and Column 3 holds 2. The cell where they cross can only be 3.
Basic
Cage Finish Basic A cage with one empty cell left: the target and the digits already in it name the last digit.
Say a 6+ cage has three cells, and the two that are filled add up to 4. The last one has to be 2.
For a − or ÷ cage there can be two answers, one bigger and one smaller than the digit already there. A 2− cage with a 3 in it could take 1 or 5; if the row or column already has a 5, it is 1.
Cage Minimums Basic In an adding cage, the smallest digits the other cells could hold cap how big each cell can be.
Take a cage that adds and pick one of its open cells. Add up the smallest digits the other open cells could hold; whatever is left of the target is the most your cell can be.
Say an 8+ cage covers three cells in one row of a 6×6. Two of them take at least 1 + 2 = 3, since cells in the same row cannot repeat, so the third is at most 5. Take the 6 note out of all three.
Cage Maximums Basic In an adding cage, the biggest digits the other cells could hold set how small each cell can be.
Take a cage that adds and pick one of its open cells. Add up the biggest digits the other open cells could hold; whatever is still missing has to come from your cell.
Say a 15+ cage covers three cells in one column of a 6×6. Two of them hold at most 6 + 5 = 11, so the third is at least 4. Take the 1, 2 and 3 notes out of all three.
Cage Divisors Basic In a multiplying cage, every open cell has to divide what is left of the target exactly.
Divide the target by the digits already in the cage. Say a three-cell 12× cage on a 5×5 already holds a 3: the other two must multiply to 4, which 3 and 5 do not divide. Take those notes out.
Intermediate
Cage Lock Intermediate A cage whose open cells can only hold one set of digits, whatever order they go in.
Say the two open cells of a 24× cage on a 6×6 can only be 4 and 6, whichever way round, since no other pair of digits up to 6 multiplies to 24. Every other note comes out of both cells.
If the two cells share a row or column, that line's 4 and 6 are spoken for, and the rest of the line loses those notes too.
Repeated Factors Intermediate An L-shaped multiplying cage whose only way to reach the target uses one digit twice.
Say an L-shaped 25× cage of three cells can only be 5 × 1 × 5. The two 5s cannot share a row or column, so they take the two ends of the L, and the 1 goes in the corner.
Cage Pairs Intermediate A two-cell − or ÷ cage with one cell filled leaves only two possible digits for the other.
Say a 2÷ cage has a 2 in one cell. The other is either twice it or half it: 4 or 1. On a 1− cage with a 4, the other is 3 or 5. Every other note comes out of the open cell.
Naked Pair Intermediate Two cells in one row or column whose notes are the same two digits.
Say two cells in Row 3 have only the notes 2 and 5. Those two digits fill those two cells, so every other cell in Row 3 loses its 2 and 5 notes.
Advanced
Cage Pointing Advanced Every way of filling a cage puts a certain digit somewhere in one row or column.
Pick a cage and a digit. Say an L-shaped cage covers two cells of Row 2 and one of Row 3, and every way of filling it puts its 6 in one of the two Row 2 cells. Row 2 gets its 6 from the cage, so every other cell of Row 2 loses its 6 note.
Cage Combinations Advanced Write out every way a cage can still be filled, and drop any note that appears in none of them.
Take the cage with the fewest open cells and list every way its digits could still go in, obeying the row and column rules. Say a three-cell 12× cage on a 6×6 could be 1, 2, 6 or 1, 3, 4, or 2, 2, 3 if the two 2s do not share a row or column.
Check each against the notes already in the cells. Any note that turns up in none of the fills that survive can come out.
