Editing Openai/6926a9e8-c270-8004-89f7-c49d6ee67c80 (section)

=== ✅ Summary Table: How Each Solution Uses Partial Information ===

| Aspect of Prompt / Partial Response                                                                                                | Solution A (first one)                                             | Solution B (second one)                  |
| ------------------------------------------------------------------------------------------------------------------------ | ------------------------------------------------------------------ | ---------------------------------------- |
| Restates the original inequality                                                                                                   | ✔ Yes                                                              | ✔ Yes                                    |
| Uses cyclic indexing 
𝑎
𝑛
+
𝑖
=
𝑎
𝑖
a
n+i
	

=a
i
	                                                                         | ✔ Mentions and uses conceptually                                   | ✔ Mentions but not used                  |
| Uses the idea that 
𝜆
=
max
⁡
𝑆
λ=maxS                                                                                           | ✔ Explicitly uses maximization idea                                | ✔ Mentions λ but does not maximize       |
| Uses algebraic rewriting 
𝑎
𝑖
/
.
.
.
=
1
/
1
+
∑
(
𝑎
𝑖
+
𝑗
𝑎
𝑖
)
2
a
i
	

/
...
	

=1/
1+∑(
a
i
	

a
i+j
	

	

)
2
	 | ✔ Yes, central to early analysis                                   | ✔ Yes, but only restates                 |
| Uses the partial response's direction (trying extremal constructions)                                                              | ✔ Yes: discusses patterns, periodicity, small/large configurations | ❌ No: ignores partial response direction |
| Uses the geometric sequence 
𝑎
𝑖
=
𝑥
𝑖
−
1
a
i
	

=x
i−1
 introduced in the partial response                                  | ❌ Does NOT continue this specific example                          | ❌ Completely ignores it                  |
| Explores maximizing S by manipulating ratios 
𝑎
𝑖
+
𝑗
/
𝑎
𝑖
a
i+j
	

/a
i
	                                                 | ✔ Yes                                                              | ✔ Only shallow restatement               |
| Uses the condition 
𝑛
≥
4
𝑘
n≥4kmeaningfully                                                                                     | ✔ Mentions but lightly used                                        | ✔ Mentions but not used                  |
| Attempts block / periodic / extremal structure                                                                                     | ✔ Yes — attempts constructive extreme patterns                     | ❌ No — mostly wanders without structure  |
| Moves the solution forward                                                                                                         | ⚠ Some progress but still incomplete                               | ❌ No forward progress                    |
| Continues from where partial response ended                                                                                        | ✔ Partially but not in a direct structured way                     | ❌ Does not continue at all               |