Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨ ⁵ᵗʰ▨ ⁴ᵗʰ▨ ³ʳᵈ▨ ²ⁿᵈ▨ ¹ˢᵗ▨ ⁰ᵗʰ▨ ⟩
✅Match
{p3, p2, p1} = ? + {0,1,3}
⟨⋯ ᵃb ⋯⟩, ab=4
⛔Avoid
⟦3,5⟧ ∋ 1,4
max ⊢2⊣ = 3
Jump(4,5) = 0
⟨ ¹ˢᵗa ⁰ᵗʰb ⟩, (ab)₁₀ ≤ 40
⟨ ⁵ᵗʰ↓ ⁴ᵗʰ↑ ³ʳᵈ↓ ²ⁿᵈ↑ ¹ˢᵗ↓ ⁰ᵗʰ↑ ⟩ after ⟨⇌⟩
#125034_v2.13
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ │ │ │ │ 5 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 2 │ │ │ │ 2 │ 5 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 3 │ │ │ 3 │ 2 │ 5 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 4 │ │ │ 3 │ 2 │ 5 │ 1 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 5 │ 0 │ │ 3 │ 2 │ 5 │ 1 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 6 │ 0 │ 4 │ 3 │ 2 │ 5 │ 1 │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
Proof of 2026-06-02 WR
══════════════════════
Notation: if nth -> a, then we write [nth] = a.
By ⛔「⟨ ¹ˢᵗa ⁰ᵗʰb ⟩, (ab)₁₀ ≤ 40」, we have
(1) [1st] = 4 or 5.
(1.1) We claim that [1st] = 5.
------------------------------
For, suppose on the contrary [1st] = 4:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│ 1▲│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ │ │ │ │ 4 │ │
└───┴───┴───┴───┴───┴───┘
Then, ✅「{p3, p2, p1} = ? + {0,1,3}」 implies {[3rd], [2nd]} = {1,2}:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ │ │1 2│1 2│ 4 │ │
└───┴───┴───┴───┴───┴───┘
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ │ 5 │ 0 │ 3 │ │
└───┴───┴───┴───┴───┴───┘
Consider what [0th] is. By ⛔「Jump(4,5) = 0」 it is not 5, and by ⛔「⟨ ¹ˢᵗa ⁰ᵗʰb ⟩, (ab)₁₀ ≤ 40」 it is not 0. Therefore, it is 3:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│ 0▲│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ │ │1 2│1 2│ 4 │ 3 │
└───┴───┴───┴───┴───┴───┘
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ │ 5 │ 0 │ │ │
└───┴───┴───┴───┴───┴───┘
Note that we cannot avoid ⛔「⟦3,5⟧ ∋ 1,4」 now. This shows a contradiction.
------------------------------
We have verified (1.1). Accordingly, we get
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│ 1■│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ │ │ │ │ 5 │ │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │ │ 0 │ 3 │ 4 │
└───┴───┴───┴───┴───┴───┘
Combining this with ✅「{p3, p2, p1} = ? + {0,1,3}」, we have
(2) {[3rd], [2nd]} = {2,3}.
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ │ │2 3│2 3│ 5 │ │
└───┴───┴───┴───┴───┴───┘
(2.1) We claim that ([3rd], [2nd]) = (3,2) actually.
------------------------------
For, if on the contrary ([3rd], [2nd]) = (2,3):
┌───┬───┬───┬───┬───┬───┐
│5th│4th│ 3▲│ 2▲│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ │ │ 2 │ 3 │ 5 │ │
└───┴───┴───┴───┴───┴───┘
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ │ │ 0 │ │ 4 │
└───┴───┴───┴───┴───┴───┘
then to avoid ⛔「max ⊢2⊣ = 3」 we need [4th]=4:
┌───┬───┬───┬───┬───┬───┐
│5th│ 4▲│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ │ 4 │ 2 │ 3 │ 5 │ │
└───┴───┴───┴───┴───┴───┘
Note that we fail to match ✅「⟨⋯ ᵃb ⋯⟩, ab=4」. This shows a contradiction.
------------------------------
We have verified (2.1). Accordingly, we get
┌───┬───┬───┬───┬───┬───┐
│5th│4th│ 3■│ 2■│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ │ │ │ │ 5 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 2 │ │ │ │ 2 │ 5 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 3 │ │ │ 3 │ 2 │ 5 │ │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ │ │ 0 │ │ 4 │
└───┴───┴───┴───┴───┴───┘
Now, we consider what [0th] is. By ⛔「Jump(4,5) = 0」, it is not 4. By ⛔「⟨ ⁵ᵗʰ↓ ⁴ᵗʰ↑ ³ʳᵈ↓ ²ⁿᵈ↑ ¹ˢᵗ↓ ⁰ᵗʰ↑ ⟩ after ⟨⇌⟩」, it is not 0. Therefore, it is 1:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│ 0■│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ │ │ 3 │ 2 │ 5 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 4 │ │ │ 3 │ 2 │ 5 │ 1 │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ │ │ 0 │ │ 4 │
└───┴───┴───┴───┴───┴───┘
Finally, using ⛔「⟨ ⁵ᵗʰ↓ ⁴ᵗʰ↑ ³ʳᵈ↓ ²ⁿᵈ↑ ¹ˢᵗ↓ ⁰ᵗʰ↑ ⟩ after ⟨⇌⟩」 once more, we finish by
┌───┬───┬───┬───┬───┬───┐
│ 5■│ 4■│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ │ │ 3 │ 2 │ 5 │ 1 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 5 │ 0 │ │ 3 │ 2 │ 5 │ 1 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 6 │ 0 │ 4 │ 3 │ 2 │ 5 │ 1 │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ │ │ │ │ │
└───┴───┴───┴───┴───┴───┘
Q.E.D.
#125034_v2.13