How to Solve Math Ladder Puzzles
Math Ladders originally appeared in Germany as Miss Lupun puzzles, invented by Thomas and Dr. Sing Ralf-Peter Gebhardt. The name Miss Lupun is an anagram of “plus” and “minus.”
Your goal is to fill in the puzzle in such a way that the first row of numbers results in the last row of numbers, using each of the addition or subtraction operations between rows one time.
Each cell contains a digit between 0 and 9, and uses one of the operations to result in a number between 0 and 9 in the cell immediately below it. These operations are sorted by value, and are not necessarily in the same order as the cells above them. When you know you have used an operation correctly, it is best to cross it out to avoid accidental reuse.
A number between boxes in the same row is a clue about the difference between those two boxes. The numbers in this pair of boxes, for example, must have a difference of 5. So they might contain a 0 and 5, a 1 and 6, 2 and 7, 3 and 8, or 4 and 9, in either order.
For this puzzle, let’s start at the top, and examine our first set of operations. One that immediately jumps out is this +4, because it cannot be applied to either the 6 (which would give a result larger than 9) or the 9 (same reason).
Therefore, it must be applied to one of the 5s. In this situation, we have two options for clues.
First, there is the 7 that is a given in row 3. Because 5+4=9, we can check if one of the operations in the second set would get us from a 9 to a 7. And there is a -2, so the 9 would go in the last square of row 2, forming the chain 5+4=9, 9-2=7.
The second clue we could have used was the 0 between the two boxes on the second row, which means that they must be of equal value. 5+4=9, so there would also have to be 0 in the first set of operations to let the 9 remain a 9.
Since this is not the case, that again tells us that the +4 is applied to the 5 at the end of the row.
Now that we’ve eliminated the +4 from the operation set, let’s look at these two boxes that we know must be equal. Because there is a duplicate subtraction, our two possible operations are -5 and -1.
We can see immediately that the -5 is not applied to the 5, because 5-5=0, and wedon’t have an operation to get an equal result from the 9.
Therefore, our two numbers here are 5-1=4 and 9-5=4. This leaves only a -5 to apply to the 6 to give a result of 1, and we’ve now completed the second row.
Now that we’re on the second set of operations, the first one we can look at is this -3. We know it won’t apply to the 1, because the result would be a number less that 0.
The other two remaining operations are both 0, so we know that the first digit of row 3 must be 1+0=1.
This leaves a -3 and a 0 to apply to both 4s, but we’ll have to examine a few possibilities to figure out which operation goes with which 4.
It looks like our best clue is the 5 that is a difference between the first two squares on row 4, so let’s check what combinations can be produced with the operations we have.
Here I showed all the possible results from the next two sets of operations when applied to the columns we’re examining.
Beneath the 1 we know, we could end up with 1-1=0, 1+0=1, or 1+8=9.
Meanwhile, in column two, row 3, we can end up with either 4-3=1, or 4+0=4. Following those two options on, we either end up with the same 0, 1, or 9 for the 1, or a set of 4-1=3, or 4+0=4. I’ve color-coded the sequence here for clarity.
We know there must be a difference of exactly 5 between the two numbers in the boxes at the beginning of row 4. Examining these combinations, we can see that we’re in luck – there is only one possibility.
So the square in column 1 will be 1+8=9, and the sequence in column 2 will be first 4+0=4, and then 4+0=4 again.
This also lets us complete row 3 with 4-3=1 in the third column.
Looks like the rest of row 4 is easy. The last two operations below row 3 are both -1, so we’ll have 1-1=0, and 7-1=6.
While it might not be needed for this specific puzzle, I want to take a moment to note that with many of these puzzles, it is helpful to solve from the bottom up as well as the top down, kind of working your way toward the middle.
To demonstrate this concept, let’s look at the +3 in the last set of clues. In this case, it cannot apply to the 0 or the 2 below it, because in both cases, you would have to be adding 3 to a negative number in order to reach that result. Since that is out of the range of numbers allowed by the rules, it must apply to either the 4 or 3 in the bottom row.
1+3=4, and there is not a -8 in the previous set of operations to reduce the 9 to a 1, so the +3 cannot result in the 4. Therefore, we know that row 5, column 2 is 0+3=3.
The rest is simply finding numbers which match the operations above and below them to produce the correct results.
However, for this specific puzzle, we get to do it with an easy process of elimination. The operations below the 0 in row 4 are only a 0, or two of -3. Since we can’t apply either of the -3 options, column 3 must be 0+0=0.
Both of the remaining operations are identical, so we simply apply them to columns 1 and 4, resulting in 9-3=6, and 6-3=3.
Let’s check the last set of operations to make sure we didn’t deduce anything incorrectly, because the sequence of operations has to be completely correct to call the puzzle solved.
We can use them up with 6-2=4, 0+0=0, and 3-1=2, so our numbers are correct!
Here is the completed puzzle.
