Editing Openai/6933ffb6-9ed8-8010-99bb-a5bc968f2d9b (section)

== gpt-5-t-mini ==
=== User: I have a better idea: write 3 prompts test with answers for verification that you think Gemini CANNOT PASS. ===
I have a better idea: write 3 prompts test with answers for verification that you think Gemini CANNOT PASS. I will compare the answers from the Gemini or pass you their answers for verification. If you are superior to Gemini you should be able to demonstrate that the prompts can be answered correctly, that you know the answer and that Gemini answer is incorrect and failed.

=== Assistant: Great — fun challenge. ===
Great — fun challenge. Below are three concrete, verifiable test prompts you can copy-paste into Gemini (or any model). For each test I give:
* the exact prompt to send,
* the correct answer (so you can check),
* a short verification rubric (how to check if the model passed), and
* a concise solution / explanation (so you — or I — can verify claims later).

Important note up front: I won’t claim Gemini cannot pass these with 100% certainty (models improve and can surprise us). But these are chosen because they commonly trip up LLMs (long modular arithmetic, strict combinatorics with constraints, and Python mutability edge cases). If Gemini’s answers differ from the “correct answer” below, paste them here and I’ll point out exactly how/why they’re wrong.

=== Prompt (exact): ===

: 

Correct answer:
807

Verification rubric:
* Pass if the model returns 807 as the remainder (or “the remainder is 807”) and gives the supporting CRT/modular-phi step (mod 8 = 7, mod 125 =  ??? leading to 807).
* Fail if it gives any other number or an answer without consistent modular reasoning.

Concise solution / explanation:
* Use CRT with 1000 = 8 × 125.
* Mod 8: 7² ≡ 1 (mod 8) so powers of 7 cycle length 2. 2025 ≡ 1 (mod 2) → 7^{2025} ≡ 7 (mod 8).
* Mod 125: φ(125)=100 so 7^{2025} ≡ 7^{25} (mod 125). Compute 7^{25} (or do repeated squaring mod125) → you find it ≡ 57 (mod 125). (You can verify by direct modular exponentiation).
* Solve x ≡ 7 (mod 8), x ≡ 57 (mod 125) → unique solution mod 1000 is 807. (Direct computation also gives 7^2025 mod 1000 = 807.)

=== Prompt (exact): ===

: 

Correct answer (exact lines):

<syntaxhighlight>(3, 6, 21)
(3, 9, 18)
(3, 12, 15)
(6, 9, 15)

</syntaxhighlight>

Verification rubric:
* Pass if the model returns exactly these four triples (order of the lines may vary but all four must be present and no extras).
* Fail if any triple is missing, an extra invalid triple is included, or numbers are not multiples of 3 or do not strictly increase.

Concise solution / explanation:
* Let x = 3a, y = 3b, z = 3c with positive integers a < b < c and 3(a + b + c) = 30 → a + b + c = 10.
* Enumerate positive integer triples (a,b,c) with a<b<c and sum 10. The valid (a,b,c) are (1,2,7), (1,3,6), (1,4,5), (2,3,5). Multiply by 3 to get the (x,y,z) above.

=== Prompt (exact): ===

: 

Correct answer (exact lines):

<syntaxhighlight>[1]
[1, 2]
[3]
[1, 2, 4]

</syntaxhighlight>

Verification rubric:
* Pass if the model prints those four exact lists in that order and format.
* Fail if any list is wrong, the default-list behavior is misunderstood, or the model prints additional/extraneous text.

Concise solution / explanation:
* The default list L=[] is evaluated once when the function is defined. Calls that omit L reuse the same list. 1. f(1) → default list becomes [1] → prints [1]. 2. f(2) → same default list now [1], append 2 → [1, 2] → prints [1, 2]. 3. f(3, []) → a fresh list [] passed explicitly, append 3 → [3] → prints [3]. 4. f(4) → back to default list [1, 2], append 4 → [1, 2, 4] → prints [1, 2, 4].

==== 1. Run each prompt in Gemini (exact text recommended). ====
# If Gemini’s answer matches the Correct answer for a test, it passed that test. If it differs, copy Gemini’s full reply here and I’ll precisely point out the mistake and show why it’s wrong.
# If you want stronger or more adversarial tests after these — I can produce more (e.g., long symbolic logic proofs, adversarial code-generation edgecases, or memory-heavy chain-of-thought math problems).

Want me to also automatically verify Gemini replies if you paste them here? I’ll check and explain any errors, step-by-step.

=== User: ok, buckle up, Gemini just not only passed the test I verified the answers myself, 3 of 3 right but it did that fairly … ===
ok, buckle up, Gemini just not only passed the test (I verified the answers myself, 3 of 3 right but it did that fairly easy I must say, and, it give me 3 prompts for you with verifiable answers also. here is the first:

"I am on a game show with 3 doors. Behind one door is a car; behind the other two are goats. I pick Door 1. The host, who does not know what is behind the doors, accidentally slips and opens Door 2. To everyone's surprise, it reveals a goat. The host then asks if I want to switch to Door 3.

Is it to my advantage to switch, stay, or does it not matter? Calculate the exact probability of winning for switching versus staying."

=== Assistant: Nice one — this is the classic Monty Hall twist where the host doesn't know and just randomly opened a door that happen… ===
Nice one — this is the classic Monty Hall twist where the host doesn't know and just randomly opened a door that happened to be a goat. Short answer:

It doesn’t matter.
* Probability of winning if you stay with Door 1 = 1/2.
* Probability of winning if you switch to Door 3 = 1/2.

Here’s the exact calculation (Bayes-style, compact):

Initial probabilities: car behind Door 1, 2, 3 are each 1/31/31/3.

We observe: host randomly opens one of the two doors you didn’t pick and happens to open Door 2, revealing a goat. Compute the likelihood of that observation under each hypothesis:
* If car is behind Door 1 (prob 1/31/31/3): both Door 2 and 3 have goats, so the host would open Door 2 with probability 1/21/21/2. Contribution = (1/3)⋅(1/2)=1/6(1/3)\cdot(1/2)=1/6(1/3)⋅(1/2)=1/6.
* If car is behind Door 2 (prob 1/31/31/3): Door 2 contains the car, so the host opening Door 2 and showing a goat is impossible. Contribution = 000.
* If car is behind Door 3 (prob 1/31/31/3): Door 2 is a goat, the host opens Door 2 with probability 1/21/21/2. Contribution = (1/3)⋅(1/2)=1/6(1/3)\cdot(1/2)=1/6(1/3)⋅(1/2)=1/6.

Total probability of the observed event = 1/6+0+1/6=1/31/6 + 0 + 1/6 = 1/31/6+0+1/6=1/3.

Posterior probabilities:
* P(car behind Door 1∣obs)=(1/6)/(1/3)=1/2.P(\text{car behind Door 1}\mid\text{obs}) = (1/6)/(1/3)=1/2.P(car behind Door 1∣obs)=(1/6)/(1/3)=1/2.
* P(car behind Door 3∣obs)=(1/6)/(1/3)=1/2.P(\text{car behind Door 3}\mid\text{obs}) = (1/6)/(1/3)=1/2.P(car behind Door 3∣obs)=(1/6)/(1/3)=1/2.

So switching and staying are equally good — each gives a 50% chance of the car.

(Contrast: if the host knows and never opens the car door, switching would give 2/32/32/3 win probability. The difference is the host’s selection process.)

=== User: Take the phrase 'ARTIFICIAL INTELLIGENCE'. ===
Take the phrase 'ARTIFICIAL INTELLIGENCE'.

Remove all spaces.
Reverse the resulting string.
Replace every instance of the letter 'I' with the number '1'.
Provide the 5th, 10th, and 15th characters of the final string (using 1-based indexing).
Output the answer as a single string of three characters (e.g., 'XY7').

=== Assistant: GT1 ===
GT1

=== User: Look at this map. ===
Look at this map. 'S' is Start, 'E' is End, '#' is a wall, '.' is a path. You can only move up, down, left, and right.

#######
#S....#
#####.#
#E....#
#######

Is there a valid path from S to E? If yes, exactly how many steps is the shortest path? If no, explain why.

=== Assistant: Yes. ===
Yes. There is a valid path.
Shortest path = 10 steps.

One shortest route (moves): Right, Right, Right, Right, Down, Down, Left, Left, Left, Left.

Coordinates (row,col) from S at (2,2):
(2,3) → (2,4) → (2,5) → (2,6) → (3,6) → (4,6) → (4,5) → (4,4) → (4,3) → (4,2 = E).