Number puzzles you can solve with logic alone
Every level in a falling number puzzle can be solved with logic alone. Here is how each step follows from the rules, with no guessing needed.
A 4x4 board has one empty cell in the top row. The three other cells in that row hold 1, 3 and 4. Faced with the gap, a solver might think any number will fit, and that the only option is to choose one and see what follows.
The row alone settles the question. Rows, columns and boxes each contain every number only once, which means the empty cell rules out 1, 3 and 4. That leaves 2 as the only candidate, and the rest of the board does not affect this.
Those who work this way check the row, then the cell's column, and then the 2x2 box it sits in. Each unit removes numbers from the list. When one number remains, it goes in. If two numbers are still possible, the solver skips that cell and checks a different one.
In this example the row alone is enough. Most boards need two or three units checked together before a cell has one answer, yet the steps stay the same and never require trying a value to see if it fails.
Checking each unit before a number drops
A solver who scans before dropping a number usually starts with one row, then moves to one column, then to one box. Every scan gives a short list of placed numbers and an even shorter list of numbers still missing. The missing numbers count for more than the placed ones, since each one still needs a spot somewhere in that unit.
A number missing from a unit can go only in an open cell of that unit. If the unit has one open cell, the number has one place to go. If it has two open cells, the solver checks the columns and boxes crossing each of them and sees if one is ruled out.
Here is a hypothetical 6x6 row with three numbers already placed, say 2, 4 and 5. Three cells are open, and the missing numbers are 1, 3 and 6. Suppose the column above one open cell already holds 1 and 3. That cell can only take 6. The row now has two open cells and two missing numbers, so the next check is shorter.
After each placement, the solver rescans the row, column and box that the new number touches. The units have changed, so a cell that had two options a moment ago may now have only one.
What the drop order does to the next move
In a game where numbers fall, each number drops to the lowest empty cell of the column the solver taps. The solver does not choose row. Each drop fixes one cell for every later move, and that cell can no longer hold any other number.
NumFall is a Puzzle game for iPhone and iPad where every level can be solved by logic alone.
Careful planners often write out two drop orders on paper before they commit to one. Picture a 4x4 column whose bottom two cells are empty and which still needs a 2 and a 4. If the 2 drops first, it lands in the bottom cell and the 4 stacks above it. Dropping the 4 first reverses that.
Next to the board, the solver writes out both versions and checks each one against the rows and boxes the two cells belong to. Imagine the bottom row already has a 2 in it. In that case the 2 cannot go there, so the 4 must drop first. Only one order breaks a rule, so the choice is made without placing any number on the board to test it.
When a level seems to need a guess
A common moment comes on a 6x6 board, when a solver finds a cell with two numbers left, say 3 and 5, and neither can be ruled out by its own row, column or box. Many solvers feel stuck at this point and assume the puzzle requires a guess. A single extra check is usually enough to resolve it.
The steps solvers often use run in this order:
- Next, search for another cell that has the same pair of options. When two cells in one row each allow only 3 and 5, those numbers are reserved for them, and no other cell in that row can use either.
- Look for a box that already contains a 3 or a 5. The number in that box cannot appear again there, so any cell in the same row or column that crosses the box loses that option.
- Skip the cell for now, place a number elsewhere, and return to it later.
The third step works since every placement alters one row, one column and one box. After a number goes in elsewhere, a cell that had two choices may be left with one. Solvers return to the same cell after every placement that touches its row, column or box, and cross out the number the new placement removes.
FAQ
How do you find the only possible number for an empty cell?
Check the row, the column and the 2x2 box that hold the cell, and cross out each number already placed in any of them. If only one number is left, it belongs in that cell. On a 4x4 board, a top row holding 1, 3 and 4 leaves only 2 for the open cell.
What should I do when a cell has two numbers left?
Look for a second cell in the same row that allows the same two numbers. If you find one, neither number can go anywhere else in that row. Otherwise, leave the cell, place a number elsewhere, and come back to it after the new placement changes its row, column or box.
Can number puzzles be solved without guessing?
Many boards can be solved by reading each unit and removing numbers that cannot fit. Players who go this way do not need to test a value and see if it causes a conflict. Some harder boards may need more than one check before a cell has a single answer.
Why does the order of dropping numbers matter in NumFall?
In NumFall, each number falls to the lowest empty cell of the column you tap, so every drop fixes a cell. Before you drop, write out both possible orders and check each one against the rows and boxes involved. The order that breaks no rule is the one to use.
NumFall is made by the publisher of this site. Any health or brain-training points here are general information, not medical advice.