2024-06-18 Q1(m=6)

Rearrange the digits in ⟨1263045⟩ to meet the rules below.

⟨6th 5th 4th 3rd 2nd 1st 0th⟩

✅Match

6th → a, 0th → b, a+b=4

5th|4th|3rd → 3

4th → a, 1st → b, ab=3+4n

⛔Avoid

5th → 2|5

min ⊢4⊣ = 4

Jump(1,6) ≥ 2

#125034_v2.5

       ┌───┬───┬───┬───┬───┬───┬───┐
       │6th│5th│4th│3rd│2nd│1st│0th│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │   │   │ 3 │   │   │   │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 2 │   │   │ 3 │   │   │ 5 │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 3 │   │   │ 3 │   │ 2 │ 5 │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 4 │ 4 │   │ 3 │   │ 2 │ 5 │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 5 │ 4 │   │ 3 │   │ 2 │ 5 │ 0 │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 6 │ 4 │ 1 │ 3 │   │ 2 │ 5 │ 0 │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 7 │ 4 │ 1 │ 3 │ 6 │ 2 │ 5 │ 0 │▒
       └───┴───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

Proof of 2024-06-18 Q1(m=6)
═══════════════════════════

Notation: if nth -> a, then we write [nth] = a.

To begin with, observe that ✅「4th → a, 1st → b, ab=3+4n」 implies the product concerned is 3 or 15. Therefore,

(1) {[4th],[1st]} = {1,3} | {3,5}.

A fortiori, 3 = [4th] | [1st]. Combining this with ✅「5th|4th|3rd → 3」, we get 3 = [4th] as our first step.

       ┌───┬───┬───┬───┬───┬───┬───┐
       │6th│5th│ 4■│3rd│2nd│1st│0th│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │   │   │ 3 │   │   │   │   │▒
       └───┴───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │ 6 │   │ 0 │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┴───┘

Then, note that (1) becomes

(2) [1st] = 1|5,

and that ✅「6th → a, 0th → b, a+b=4」 implies

(3) {[6th],[0th]} = {0,4}.

┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
│ ▬ │   │ 3 │   │   │   │ ▬ │
└───┴───┴───┴───┴───┴───┴───┘

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │ 6 │   │   │   │ 5 │
└───┴───┴───┴───┴───┴───┴───┘

It follows that [5th] = 1|2|5|6. In view of ⛔「5th → 2|5」, actually it is [5th] = 1|6. On the other hand, by ⛔「Jump(1,6) ≥ 2」, there is at most one digit between 1 and 6. Therefore, we have {[5th], [3rd]} = {1,6}:

┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
│ ▬ │ / │ 3 │ / │   │   │ ▬ │
└───┴───┴───┴───┴───┴───┴───┘

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│   │ 2 │   │   │   │   │ 5 │
└───┴───┴───┴───┴───┴───┴───┘

It follows that [1st] = 2|5. In view of (2), we have [1st] = 5 indeed. As a result, we get

       ┌───┬───┬───┬───┬───┬───┬───┐
       │6th│5th│4th│3rd│ 2■│ 1■│0th│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
       │   │   │ 3 │   │   │   │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 2 │   │   │ 3 │   │   │ 5 │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 3 │   │   │ 3 │   │ 2 │ 5 │   │▒
       └───┴───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ 1 │   │ 6 │   │ 0 │ 4 │   │
└───┴───┴───┴───┴───┴───┴───┘

Next, note that to match (3) and avoid ⛔「min ⊢4⊣ = 4」 at the same time, we need:

       ┌───┬───┬───┬───┬───┬───┬───┐
       │ 6■│5th│4th│3rd│2nd│1st│ 0■│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
       │   │   │ 3 │   │ 2 │ 5 │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 4 │ 4 │   │ 3 │   │ 2 │ 5 │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 5 │ 4 │   │ 3 │   │ 2 │ 5 │ 0 │▒
       └───┴───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ 1 │   │ 6 │   │   │   │   │
└───┴───┴───┴───┴───┴───┴───┘

Finally, using ⛔「min ⊢4⊣ = 4」 once more, we finish by

       ┌───┬───┬───┬───┬───┬───┬───┐
       │6th│ 5■│4th│ 3■│2nd│1st│0th│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
       │ 4 │   │ 3 │   │ 2 │ 5 │ 0 │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 6 │ 4 │ 1 │ 3 │   │ 2 │ 5 │ 0 │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 7 │ 4 │ 1 │ 3 │ 6 │ 2 │ 5 │ 0 │▒
       └───┴───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│   │   │   │   │   │   │   │
└───┴───┴───┴───┴───┴───┴───┘

Q.E.D.

#125034_v2.5