# 125034 v1.0.0 / WR 1713-11-07

I am developing an app that generates puzzle games every day (https://125034.pages.dev/). Their basic form may be viewed as Wordle in math, Sudoku in 6 grids, or Einstein's Riddle in 1 row.

If today was 1713-11-07 and the app version was v1.0.0, it would produce the following puzzle:

WR 1713-11-07

Rearrange the digits in ⟨125034⟩ to meet the rules below.

⟨5th 4th 3rd 2nd 1st 0th⟩

✅Match
⟨⋯ 2 ⋯ ? 3 ⋯ (?+5)⟩ (?≠3)

⛔Avoid
1st → a, 0th → b, |a-b|=1
4th → 1|4|5
3rd → a, 0th → b, a+b=2+5n
⟨⋯ 4 ⋯ 1 ⋯⟩
3rd → a, 2nd → b, |a-b|=1

#125034 v1.0.0


In other words, we are asked to find a permutation of ⟨125034⟩ so that all of the required patterns are matched and all of the forbidden patterns are avoided.

What is the answer and how do you approach it? Imagine you are playing Wordle in which each right digit gives you a green box while each wrong digit gives you a red box. To have a solution that you can share with delight, perhaps your steps may only exhibit an increasing no. of green boxes without showing any red boxes.

The "match" pattern gives a lot of info - the only two digits with a difference of 5 are 0 and 5. This means that the pattern must take the form *2*03*5.

The fourth "avoid" pattern says that 1 must come before 4. The first "avoid" pattern implies that the ending can't be "45". This means the rightmost * can't be occupied, and we're down to only 3 options: 142035, 124035, 214035. The second "avoid" rule tells us that it's the second of those options, 124035.

I solved this puzzle with , as requested. But if I had the choice, I absolutely would use computers to solve this puzzle, and any others of the type, rather than attempt them manually. In general, for computer-generated puzzles there might not be a 'clean' logical path; I wouldn't expect one except coincidentally. So it would be much easier to just generate and filter all 720 permutations in a spreadsheet, rather than attempting to search for a logical path that will probably be uninteresting or nonexistent.

And this puzzle is very clearly computer-generated. Multiple clues are entirely redundant, and of the other clues, some parts of them are absolutely useless (why state that the 4th digit isn't a 5, when we already know that 5 is the 0th digit?).

The hardest part of it was parsing the syntax - the clues are written in a very artificial way. Why label the digits of the answer in reverse order? Why start from 0? Why use variable assignments in the clues? These might make sense for efficiency in generating the puzzles, but they definitely don't make solving easier.

• I am the op and thank you very much for your comments. It gives me some meaningful reflection and I really like it. Let's back to the puzzle itself, though it may be imperfect and confusing. I think your answer is interesting, but firstly, my intention is that the "match" pattern allows that the first digit can be 2, so e.g. 203145 is possible. Also, your approach may be slightly computer-like (reflecting what you think of the puzzle?), while I have at least two logical solutions to the puzzle, and I am interested in seeing more logical deduction. Dec 22, 2022 at 4:00
• And about the rules you think redundant, I think we can start by using any rules for the logical deduction. If we do not start by the "match" pattern, then we don't know that some part of the "avoid" pattern 2 is redundant. I have included the tag combinatorics, because players can judge by combinatorics which rules they should use as the initial step. Dec 22, 2022 at 4:18
• And about your comments on syntax, I have included the tag mathematics, because when we mention 125034 as a number in the Arabic numeral system (a positional system), this number is actually 1*10^5 + 2*10^4 + 5*10^3 + 0*10^2 + 3*10^1 + 4*10^0. Dec 22, 2022 at 4:32
• Yes, I realized that the "match" allows the first digit to be 2; that's why there was a star at the start of *2*03*5. I included one option starting with 2 when I had narrowed it down to 3. As for logical deduction, my answer was perfectly logical, wasn't it? I made deductions, determining certain aspects of the pattern until I reached the solution. What makes it not "logical"? This was my natural approach to the puzzle, and the approach I took didn't seem in any way 'artificial' to me.
– Deusovi
Dec 22, 2022 at 8:20
• As for tags, I don't see any particular combinatorial principle that you expect people to use. I recognize that some amount of combinatorics can be used to describe the set of permutations of {0,1,2,3,4,5}, but that doesn't actually factor into the solve. And [mathematics] doesn't seem very relevant either - by that standard, it would be a tag for any puzzle involving a string of digits! In both of these cases, the tag doesn't seem very helpful for describing the question.
– Deusovi
Dec 22, 2022 at 8:22

I am the op and I would like to share one approach too. It can be summarized as:

WR 1713-11-07

⚫⚫⚫⚫⚫✅
⚫✅⚫⚫⚫✅
✅✅⚫⚫⚫✅
✅✅⚫✅⚫✅
✅✅⚫✅✅✅
✅✅✅✅✅✅

#125034 v1.0.0


(It may be interesting to try to reproduce my deduction just from this diagram)

The details are as follows.

Step 1:

By ✅⟨⋯ 2 ⋯ ? 3 ⋯ (?+5)⟩ (?≠3), we see that ? has to be 0, otherwise (?+5) would be out of bound. Hence, we get:
⚫⚫⚫⚫⚫✅
⟨*****5⟩.
Here we use * as a placeholder for any unsolved position.

Step 2:

By ⛔4th → 1|4|5, we see that the 4th position can only be 0,2,3.
If it is 0, then by ✅⟨⋯ 2 ⋯ ? 3 ⋯ (?+5)⟩ (?≠3) with ?=0 (step 1), we get ⟨203**5⟩. And then ⛔⟨⋯ 4 ⋯ 1 ⋯⟩ forces ⟨203145⟩. It matches ⛔1st → a, 0th → b, |a-b|=1 however, which is a contradiction.

Else if the 4th position is 3, then by ✅⟨⋯ 2 ⋯ ? 3 ⋯ (?+5)⟩ (?≠3) with ?=0 (step 1), we get ⟨03***5⟩, leaving no place for 2. So again it is a contradiction. Therefore, the 4th position is 2:
⚫✅⚫⚫⚫✅
⟨*2***5⟩.

Step 3:

By ✅⟨⋯ 2 ⋯ ? 3 ⋯ (?+5)⟩ (?≠3) with ?=0 (step 1), we can treat 03 as a single block. This block can only be filled in the following two ways: ⟨*203*5⟩ or ⟨*2*035⟩. In any case, to submit the remaining digits, namely 1,4, we have to avoid ⛔⟨⋯ 4 ⋯ 1 ⋯⟩. So, 1 is the leading digit:
✅✅⚫⚫⚫✅
⟨12***5⟩.

Step 4:

To avoid ⛔1st → a, 0th → b, |a-b|=1, the 1st position cannot be 4. Therefore, the block 03 from step 3 has to be filled this way:
✅✅⚫✅✅✅
⟨12*035⟩.

Step 5:

We have digit 4 remains. Submitting it, we reach
✅✅✅✅✅✅
⟨124035⟩!

We remark that

We have not used the following rules:
⛔3rd → a, 0th → b, a+b=2+5n
⛔3rd → a, 2nd → b, |a-b|=1.

It is only a developer notes:

Redundant rules were included in v1.0.0 for reducing the game's difficulty so as to reach wider audience. However, it seems that elite players do not like any of them. So, they are avoided in the current v1.1.

Finally, I thank the players of the games in this holiday season. Unfortunately it is an imperfect puzzle which receives a number of downvotes. Please feel free to give me a comment if you would like to share why you don't like it. Wish you a merry christmas!

Assuming that the ellipses represent any number of terms (including none):

✅Match: ⟨⋯ 2 ⋯ ? 3 ⋯ (?+5)⟩ (?≠3):

As we are permuting digits 0-5, the ? and (?+5) can only be 0 and 5. The ?≠3 requirement appears to be redundant.
So we have 2,0,3,5 in that order, but not necessarily adjacent, with the 1 and 4 inserted somewhere.

⛔Avoid: ⟨⋯ 4 ⋯ 1 ⋯⟩:

This tells us the 1 must come before the 4.
There are now only ten possibilities, so let's list them before we go any further:
1 4 2 0 3 5
1 2 4 0 3 5
1 2 0 3 4 5
1 2 0 3 5 4
2 1 4 0 3 5
2 1 0 3 4 5
2 1 0 3 5 4
2 0 3 1 4 5
2 0 3 1 5 4
2 0 3 5 1 4

⛔Avoid: 1st → a, 0th → b, |a-b|=1:

The last two digits (1st and 0th) must not differ by one.
This eliminates all options that end 4,5 or 5,4.
Leaving:
1 4 2 0 3 5
1 2 4 0 3 5
2 1 4 0 3 5
2 0 3 5 1 4

⛔Avoid: 4th → 1|4|5:

The second (4th) digit cannot be 1 or 4 or 5.
Eliminates two further options, leaving just two:
1 2 4 0 3 5
2 0 3 5 1 4

⛔Avoid: 3rd → a, 2nd → b, |a-b|=1:

The third and fourth digits (3rd and 2nd) must not differ by one.
Redundant.

⛔Avoid: 3rd → a, 0th → b, a+b=2+5n

Just check this for both remaining options:
1 2 4 0 3 5 : 4+5 = 9 ≠ 2 mod 5
2 0 3 5 1 4 : 3+4 = 7 = 2 mod 5