Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟨⋯ 3 ⋯ 4 ⋯ 5 ⋯⟩
max ⊢0⊣ ≤ 4
3rd → a, 0th → b, ab=0+4n
⛔Avoid
⟨⋯ ? ⋯ 1 ⋯ (?−1)⟩ (?≠1,2)
Sim⟨ ⁵ᵗʰ3 ⁴ᵗʰ5 ³ʳᵈ1 ²ⁿᵈ0 ¹ˢᵗ4 ⁰ᵗʰ2 ⟩ ≤ 1
5th → a, 1st → b, |a-b|=1
#125034_v2.8
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ 3 │ │ │ │ │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 2 │ 3 │ │ │ 0 │ │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 3 │ 3 │ │ 4 │ 0 │ │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 4 │ 3 │ │ 4 │ 0 │ │ 5 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 5 │ 3 │ 2 │ 4 │ 0 │ │ 5 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 6 │ 3 │ 2 │ 4 │ 0 │ 1 │ 5 │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
Proof of 2025-03-18 WR
══════════════════════
Notation: if nth -> a, then we write [nth] = a.
Observe that by combining ✅「3rd → a, 0th → b, ab=0+4n」 with ✅「⟨⋯ 3 ⋯ 4 ⋯ 5 ⋯⟩」, we have
(1) one of the following holds:
(a) 0 = [3rd];
(b) 0 = [0th];
(c) 4 = [3rd].
(2) From this, we claim that 3 = [5th].
------------------------------
If on the contrary (2) does not hold, then by ✅「⟨⋯ 3 ⋯ 4 ⋯ 5 ⋯⟩」, there are three possible positions for 3:
┌───┬───┬───┬───┬───┬───┐
│5th│ 4▲│ 3▲│ 2▲│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
(2.1) │ │ │ │ 3 │ 4 │ 5 │
├───┼───┼───┼───┼───┼───┤
(2.2) │ │ │ 3 │ │ │ │
├───┼───┼───┼───┼───┼───┤
(2.3) │ │ 3 │ │ │ │ │
└───┴───┴───┴───┴───┴───┘
We show that all of them give contradictions. The following will be used frequently: To avoid ⛔「Sim⟨ ⁵ᵗʰ3 ⁴ᵗʰ5 ³ʳᵈ1 ²ⁿᵈ0 ¹ˢᵗ4 ⁰ᵗʰ2 ⟩ ≤ 1」, we need at least two agreed positional digits with ⟨351042⟩.
(3) If (2.1) holds, then by (1), we have
┌───┬───┬───┬───┬───┬───┐
│5th│4th│ 3▲│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ │ │ 0 │ 3 │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┘
However, we will fail to have two agreed positional digits with ⟨351042⟩.
(4) Else if (2.2) holds, then by (1), we have
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│ 0▲│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ │ │ 3 │ │ │ 0 │
└───┴───┴───┴───┴───┴───┘
Using ✅「⟨⋯ 3 ⋯ 4 ⋯ 5 ⋯⟩」, we get
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│ 2▲│ 1▲│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ │ │ 3 │ 4 │ 5 │ 0 │
└───┴───┴───┴───┴───┴───┘
But again, we fail to have two agreed positional digits with ⟨351042⟩.
(5) Lastly, Suppose (2.3) holds:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ │ 3 │ │ │ │ │
└───┴───┴───┴───┴───┴───┘
By (1), it follows three cases:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
(5.1) │ │ 3 │ 0 │ │ │ │
├───┼───┼───┼───┼───┼───┤
(5.2) │ │ 3 │ │ │ │ 0 │
├───┼───┼───┼───┼───┼───┤
(5.3) │ │ 3 │ 4 │ │ │ │
└───┴───┴───┴───┴───┴───┘
To have at least two agreed positional digits with ⟨351042⟩, (5.1) and (5.2) become
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ │ 3 │ 0 │ │ 4 │ 2 │
├───┼───┼───┼───┼───┼───┤
│ │ 3 │ 1 │ │ 4 │ 0 │
└───┴───┴───┴───┴───┴───┘
In both cases we cannot match ✅「⟨⋯ 3 ⋯ 4 ⋯ 5 ⋯⟩」. Therefore, (5.1) and (5.2) do not hold, and (5.3) holds instead:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ │ 3 │ 4 │ │ │ │
└───┴───┴───┴───┴───┴───┘
Then, to have at least two agreed positional digits with ⟨351042⟩, we need
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│ 2▲│1st│ 0▲│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ │ 3 │ 4 │ 0 │ │ 2 │
└───┴───┴───┴───┴───┴───┘
and to match ✅「⟨⋯ 3 ⋯ 4 ⋯ 5 ⋯⟩」, we reach
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│ 1▲│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ │ 3 │ 4 │ 0 │ 5 │ 2 │
└───┴───┴───┴───┴───┴───┘
Now we fail to match ✅「max ⊢0⊣ ≤ 4」, which is a contradiction.
------------------------------
We have verified (2). Accordingly, we get
┌───┬───┬───┬───┬───┬───┐
│ 5■│4th│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ 3 │ │ │ │ │ │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │ 5 │ 0 │ │ 4 │
└───┴───┴───┴───┴───┴───┘
Next, we consider how to have one more agreed positional digit with ⟨351042⟩.
Observe that this digit cannot be
# 5 (by ✅「⟨⋯ 3 ⋯ 4 ⋯ 5 ⋯⟩」)
# 4 (by ⛔「5th → a, 1st → b, |a-b|=1」)
# 2 (by ⛔「⟨⋯ ? ⋯ 1 ⋯ (?−1)⟩ (?≠1,2)」).
Therefore, one of the following holds:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│ 3▲│ 2▲│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
(6) │ 3 │ │ 1 │ │ │ │
├───┼───┼───┼───┼───┼───┤
(7) │ 3 │ │ │ 0 │ │ │
└───┴───┴───┴───┴───┴───┘
(8) We show that (7) holds actually.
------------------------------
For, if (6) holds, then it follows from (1) that
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│ 0▲│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ 3 │ │ 1 │ │ │ 0 │
└───┴───┴───┴───┴───┴───┘
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ 2 │ 5 │ │ │ 4 │
└───┴───┴───┴───┴───┴───┘
And to avoid ⛔「5th → a, 1st → b, |a-b|=1」, we have
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│ 1▲│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ 3 │ │ 1 │ │ 5 │ 0 │
└───┴───┴───┴───┴───┴───┘
Note that it fails to match ✅「max ⊢0⊣ ≤ 4」, which is a contradiction.
------------------------------
We have verified (8). So we have
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│ 2■│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ 3 │ │ │ │ │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 2 │ 3 │ │ │ 0 │ │ │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │ 5 │ │ │ 4 │
└───┴───┴───┴───┴───┴───┘
Plainly, (1) implies
┌───┬───┬───┬───┬───┬───┐
│5th│4th│ 3■│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ 3 │ │ │ 0 │ │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 3 │ 3 │ │ 4 │ 0 │ │ │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │ 5 │ │ │ │
└───┴───┴───┴───┴───┴───┘
We consider how to place 5. To match ✅「⟨⋯ 3 ⋯ 4 ⋯ 5 ⋯⟩」 and ✅「max ⊢0⊣ ≤ 4」 at the same time, we have
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│ 0■│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ 3 │ │ 4 │ 0 │ │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 4 │ 3 │ │ 4 │ 0 │ │ 5 │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │ │ │ │ │
└───┴───┴───┴───┴───┴───┘
Finally, to avoid ⛔「5th → a, 1st → b, |a-b|=1」, we finish by
┌───┬───┬───┬───┬───┬───┐
│5th│ 4■│3rd│2nd│ 1■│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ 3 │ │ 4 │ 0 │ │ 5 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 5 │ 3 │ 2 │ 4 │ 0 │ │ 5 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 6 │ 3 │ 2 │ 4 │ 0 │ 1 │ 5 │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ │ │ │ │ │
└───┴───┴───┴───┴───┴───┘
Q.E.D.
#125034_v2.8