Congrats to Carol.O on FTF!
Puzzle:
You can get the coordinates by solving the following: The nine letters (A, B, C, D, E, F, G, H, J) represent the digits from 1 to 9. No two letters are equal.
A * F = H
A * J = B
D - A = E
F + C = D
The final coordinates are at N 45 04.X, W 064 25.Y , where you can calculate X and Y as:
X=(A*G*D)+(H*B)+B+C
Y=(B*D)+(E*J*A)+D+(C*F)-G-H
You can check your answers for this puzzle on GeoChecker.com.
Tutorial:
This is not the only way to solve them, but it's one way that works. If you have different ways you like to solve this kind of puzzle, please feel free to post them in your log or send me a message.
To solve a logic puzzle like this, you need to use the clues provided to eliminate possible values for each of the letters. I find it helpful to use a table to keep track of possible values. I start with a blank table, and mark an "X" at each combination that is eliminated.
As 4CeasonS points out in a note, making a table like the ones below in a spreadsheet is handy. Additionally, you can use the spreadsheet to check whether your guesses are possible, by having cells for the values of the letters and cells that check the formulas using the values. The "Undo" option of a spreadsheet makes it easy to "back up" when you have made an assumption that turns out to be incorrect.
Look at the first formula: A * F = H. It turns out that there are just two possibilities: 2 * 3 = 6 and 2 * 4 = 8.
Here's why these are the only two possible values in the formulas: If you look at possible values of H, there are no other values between 1 and 9 that are a product of two other different numbers from 1 to 9. For example 1 * 2 = 2 is the only way to get 2, and you can't use that because there are two "2"s.
What does this tell us? First, you can see that H is either 6 or 8. So we can eliminate all other values for H:
We know that A and F can only be 2, 3, or 4. So we can eliminate the values 1 and 5-9 in the "A" and "F" columns:
Since there are only two possible combinations for the A*F=H equation (2*3=6 or 2*4=8), we know that one of A or F must be equal to "2", and none of the other letters can be equal to 2:
The second formula A * J = B tells us that B must be 6 or 8, and J must be 2, 3, or 4. And we already know that J is not equal to 2:
Since there are two formulas A*F=H and A*J=B, we know that one must be 2*3=6 and the other must be 2*4=8. This means that A, F and J must be some combination of 2, 3 or 4 and none of the other letters could be a 2, 3, or 4:
The only number in both of the first two formulas is "2", and the only letter is "A". So we must have A=2:
Looking at the formula D - A = E, we can see that this is the same as D=A+E. Since it is the sum of two numbers, D can't be 1. And E can't be 8 or 9 since D is a single digit and D = A+E = 2 + E.
Looking at the formula F + C = D, we can argue that F is not 3. This is because already know F = 3 or 4. So C cannot be 7, 8 or 9, since these would make D larger than 9.
Sometimes, there isn't an obvious case you can eliminate. This might be such a time. One way forward is to guess at a letter, and then continue. If you later reach a contradiction, then you have to backtrack to where you made that guess, and try another guess. If you're doing this with a spreadsheet, then backtracking is straightforward.
Here, we're going to guess that F=4. Then J=3 (using the formulas A*F=H and A*J=B) and we get this:
And now that we know F=4, we have H = 2*4=8 and J=3 gives us B=2*3=6:
C is either 1 or 5. I'm going to guess 5:
This leads us to D=9 (since F+C=D, or 4+5=9). And now we can also fill in the rest of the grid easily. We finish with a grid that has no contradictions, so our two guesses were lucky, and we have a final solution.
Although we have a solution in this puzzle, there are a few other tricks that may come in handy:
- Final coordinate formula: Sometimes there are one or more additional formulas used to convert the solution into the final coordinates. Usually these are numbers between 000 and 999, corresponding to the decimal minutes of the coordinates. Depending on the way these additional formulas are set up, they might help you eliminate possibilities. Some values of A-J might give you coordinates that are outside the range 000 - 999, and so could be eliminated.
- [Aug 20] If the final coordinate formula gives each digit individually (e.g. N 45 04.UVW; W 064 25.XYZ), then you may be able to use the fact that each of U - Z must be between 1 and 9 to eliminate some values for A-J .
- More tips to be added as I think of them....
Solving the puzzle with a computer:
The use of a spreadsheet mentioned above is one way you can use a computer, to keep track of a grid of guesses and check your math.
Another way to use a computer is to write programs that list all possible values, and then use the formulas to eliminate the cases that are impossible. In order to do this approach, you need to be familiar with writing computer programs or using programs that can manipulate data. I typically do this to solve micro logics, and have to modify my code for every puzzle. It's beyond the scope of this tutorial to provide such code, but here's a summary of the results: