=====Walkthrough===== Following will be a walkthrough (the documentation in a form acceptable for submission, provided as an example of both the form I'd like to see submissions in, but also providing illumination in the process of solving a letter division puzzle) of the following puzzle: GLJK +--------- KJKK | GLMBRVLR -VKOKL ===== LJBGV -OKVKG ===== JJGKL -LKBKV ===== KVRMR -JKRKB ===== VKMK base: 10 letters: BGJKLMOPRV The intent of the documentation (and therefore the project you are performing) is NOT merely to SOLVE the problem, but to document the step-by-step process of solving the problem. That is demonstrated here. ====Getting underway==== Right from the start, we can already make some important connections; looking at EACH of the subtractions taking place, in the left-most position, we see an interesting phenomenon taking place- G-V=0, L-O=0, J-L=0, and K-J=0. Now, since EACH letter is its own unique numeric value, subtracting one letter from another on its own won't result in a value of 0, but being borrowed from will. That is: 7-6=1, but (7-1)-6=0. THAT is what is going on here. So what we can infer from this, is some very important connections: V is one less than G (I'll write it as: V < G) O is one less than L (O < L) L is one less than J (L < J) J is one less than K (J < K) Does that make sense? From looking at the puzzle, those four relations can be made. Now, FURTHERMORE, some of those connections are thereby connected. Look at the 'L' and 'J' connections: O < L, but also: L < J L < J, but also: J < K That implies a further connection, so we can chain them together: O < L < J < K So from that initial observation and connection, we now have two disconnected relationships: V < G O < L < J < K From what we've done so far, we do not know where V,G fall in respect to O,L,J,K. They might be less than, OR greater than. We won't know without further information. Yet, even WITH this information, we can update our letter ranges: * since V is less than G, we know V can NOT be 9. * similarly, G can NOT be 0. O cannot be 9, 8, 7, because we know O is 3 less than K. So even though we don't know what K actually is, because K COULD be 9, we know what O, L, and J can NOT be. * L cannot be 9 or 8 * J cannot be 9 on the other side: * K cannot be 0, 1, or 2 * J cannot be 0 or 1 * L cannot be 0. So, if we update our range chart accordingly: B = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 } G = { 1, 2, 3, 4, 5, 6, 7, 8, 9 } J = { 2, 3, 4, 5, 6, 7, 8, } K = { 3, 4, 5, 6, 7, 8, 9 } L = { 1, 2, 3, 4, 5, 6, 7, } M = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 } O = { 0, 1, 2, 3, 4, 5, 6, } P = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 } R = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 } V = { 0, 1, 2, 3, 4, 5, 6, 7, 8 } Moving on, dealing with details of discovering those one-off relations, that tells us something about the NEXT subtractions: that they borrow (which means they are LESS THAN the thing being subtracted from them): * L is less than K (which we actually know to be 2 less than K), so L - K needs to BORROW * J is less than K (which we know is 1 less than K), so J - K needs to BORROW * V is apparently also less than K (which we didn't previously know), so V - K needs to BORROW now knowing than V << K, we can connect our other relational fragment in (I use the double '<<' to denote "less than" by an unknown amount, because while we know V is less than K, we don't know by how much). So: V < G << O < L < J < K This allows us some further whittling of our ranges: V cannot be 9, 8, 7, 6, or 5 G cannot be 9, 8, 7, or 6 O cannot be 0, or 1 L cannot be 0, 1, or 2 J cannot be 0, 1, 2, or 3 K cannot be 0, 1, 2, 3, or 4 B = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 } G = { 1, 2, 3, 4, 5, } J = { 4, 5, 6, 7, 8, } K = { 5, 6, 7, 8, 9 } L = { 3, 4, 5, 6, 7, } M = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 } O = { 2, 3, 4, 5, 6, } P = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 } R = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 } V = { 0, 1, 2, 3, 4, } Already we can see that V and G are likely lower numbers, and O, L, J, and K are likely higher numbers. What else do we have? Let's keep going: We cannot instantly proceed to the next subtraction in as obvious a progression, as we'll need more information on the various letters involved. ====Finding K (and J and L and O as well)==== However, looking at the puzzle, I'm interested in seeing if we can find any obvious examples of 0. You know, letter minus same letter sort of things. Because they will typically end up equalling 0 (or 9). Why 9? Because of a borrow! ((5-1)+10)-5 = (4+10)-5 = 14 - 5 = 9 … that can be quite revealing too! And it would appear we have one wonderful candidate in the bottom-most subtraction: KVRMR -JKRKB ===== VKMK Lookie there: R-R = K. Usually, that would result in a 0. BUT, we also know that K can NOT be 0 (looking at our range table above). So, that means it is being borrowed from, and it itself has to borrow, so we now also know that M is less than K (M << K). And, as indicated above: ((R-1)+10)-R = 9! We now know that K = 9! That suddenly reveals a whole lot to us, due to our relational chains we've built. Let's update: 0 1 2 3 4 5 6 O 7 L 8 J 9 K Also, with the new introduction of M being less than K: B = { 0, 1, 2, 3, 4, 5, } G = { 1, 2, 3, 4, 5, } J = { 8 } K = { 9 } L = { 7 } M = { 0, 1, 2, 3, 4, 5, } O = { 6 } P = { 0, 1, 2, 3, 4, 5, } R = { 0, 1, 2, 3, 4, 5, } V = { 0, 1, 2, 3, 4, } And, our relational chains: V < G << O < L < J < K M << O < L < J < K Because we don't yet know any relation of M compared to V or G, we have to keep them separate for now. We also have a second disqualifier for K being 0... the ones place subtraction in that bottom-most subtraction: R - B = K. There's nothing further to the right that could borrow from this problem, so it can only exist in two states: R is greater than B R is less than B Since we know that K is 9, there's NO OTHER pair of single digit numbers we can subtract to get 9, which tells us that: R is less than B (R << B) Currently both R and B can be 0-5 (although now, B is 1-5, and R is 0-4). We'd need to find a combination where (R+10)-B is 9: R: 0 R: 1 R: 2 R: 3 R: 4 (0+10) (1+10) (2+10) (3+10) (4+10) 10 11 12 13 14 And from that, we're subtracting B, which is 1, 2, 3, 4, or 5. The answer has to be 9. So: 10-1=9, 11-2=9, 12-3=9, 13-4=9, and 14-5=9 Hey, look at that… B is one greater than R (not just R << B, BUT: R < B) Our relational chains: V < G << O < L < J < K M << O < L < J < K R < B << O < L < J < K And our chart, of sorts: B = { 1, 2, 3, 4, 5, } G = { 1, 2, 3, 4, 5, } J = { 8 } K = { 9 } L = { 7 } M = { 0, 1, 2, 3, 4, 5, } O = { 6 } P = { 0, 1, 2, 3, 4, 5, } R = { 0, 1, 2, 3, 4, } V = { 0, 1, 2, 3, 4, } If you look, the only letter we've not yet directly interacted with yet is 'P', although we already know enough about it (that it is 0-5, less than O, L, J, and K). And if you look closely, you'll notice that 'P' isn't even present in the letter division problem! So its identity will rely entirely on the proving of the other values. Let's continue on: M-K=M, BECAUSE we know M << K, AND BECAUSE we know the subtraction to the right is borrowing from it (because R < B), we have something like this: (M-1+10)-K=M Can't really do much more with it at this point, but it is important to know to help us identify the borrows needing to happen. ====Finding our zero value (R and B)==== Why don't we go ahead and find 0? If you look in the subtraction above the bottom one, we have another "letter minus same letter" scenario, and it doesn't equal K! JJGKL -LKBKV ===== KVRM We KNOW that V << L, so no borrow is happening there. Therefore, K-K, or 9-9, equals 0. So R is 0! … and B is 1! Because of our identified relationship. Updating things! 0 R 1 B 2 3 4 5 6 O 7 L 8 J 9 K Also, with the new introduction of M being less than K: B = { 1 } G = { 3, 4, 5, } J = { 8 } K = { 9 } L = { 7 } M = { 2, 3, 4, 5, } O = { 6 } P = { 2, 3, 4, 5, } R = { 0 } V = { 2, 3, 4, } *NOTE: G is NOT 2, because G is greater than V (one greater, in fact), so we can similarly whittle that off. Relational chains can look as follows now: R < B << V < G << O < L < J < K R < B << M << O < L < J < K R < B << P << O < L < J < K Basically just down to V, G, P, and M. ====Finding V and G==== And I think we have the means to find V: notice the second to last subtraction, the "LKBKV". You know where we get that from? Multiplying the divisor (KJKK) by J (since it is the third subtraction taking place). We KNOW the numeric values of K and J, in fact we know the values of L, K, and B. The only thing we don't know is 'V', and since V is in the one's place, that makes things super easy for us. KJKK = 9899 J = 8 So: 9899 x 8 = 79192 = LKBKV! V is 2! Which means, because V < G, that G is 3! Updating our records: 0 R 1 B 2 V 3 G 4 5 6 O 7 L 8 J 9 K Also, with the new introduction of M being less than K: B = { 1 } G = { 3 } J = { 8 } K = { 9 } L = { 7 } M = { 4, 5, } O = { 6 } P = { 4, 5, } R = { 0 } V = { 2 } Relational chains can look as follows now: R < B < V < G << M << O < L < J < K R < B < V < G << P << O < L < J < K ====Finding M and discovering P==== And then there were 2. We really just need to find M, or P, and we're done. And since there are no 'P' values in the puzzle, we need to target M. So let's look for some candidates: Hey, how about this: JJGKL -LKBKV ===== KVRM One's place subtraction: L - V = M. We KNOW L (7) is greater than V (2), so no borrow is happening. L-V=M 7-2=5 M is 5. That means P is 4 by process of elimination. Puzzle completed: 0 R 1 B 2 V 3 G 4 P 5 M 6 O 7 L 8 J 9 K Also, with the new introduction of M being less than K: B = { 1 } G = { 3 } J = { 8 } K = { 9 } L = { 7 } M = { 5 } O = { 6 } P = { 4 } R = { 0 } V = { 2 } Relational chains can look as follows now: R < B < V < G < P < M < O < L < J < K Solution is obtained! Onto verification: ====Verifying our key==== The best way to verify the puzzle with our key is to convert the dividend and divisor to its numeric equivalent, perform the division, and compare the resulting quotient and remainder against those found in the letterified puzzle: divisor: KJKK --> 9899 dividend: GLMBRVLR --> 37510270 And let's do some long division! +--------- 9899 | 37510270 9899 goes into 37510 three times: 3 +--------- 9899 | 37510270 -29697 ===== 78132 It might be convenient to have a quick factor reference for 9899 handy: 9899 * 0 = 0 9899 * 1 = 9899 9899 * 2 = 19798 9899 * 3 = 29697 9899 * 4 = 39596 9899 * 5 = 49495 9899 * 6 = 59394 9899 * 7 = 69293 9899 * 8 = 79192 9899 * 9 = 89091 9899 fits into 78132 seven times (69293): 37 +--------- 9899 | 37510270 -29697 ===== 78132 -69293 ===== 88397 Once again, looking at the list of factors, we see that the best fit for 9899 into 88397 is 79192 (a factor of 8): 378 +--------- 9899 | 37510270 -29697 ===== 78132 -69293 ===== 88397 -79192 ===== 92050 Finally, a factor of 9 (89091) fits in best: 3789 <-- quotient +--------- 9899 | 37510270 -29697 ===== 78132 -69293 ===== 88397 -79192 ===== 92050 -89091 ===== 2959 <-- remainder Converting our quotient and remainder back to letters: * quotient: 3789 --> GLJK * remainder: 2959 --> VKMK And comparing against the problem we were given: * quotient: GLJK <-> GLJK * remainder: VKMK <-> VKMK Success!