六道語言模型自信卻答錯的題，與引擎算出的結果並列。
靠模式作答的助理，出錯的頻率足以造成影響，而且錯得有規律：答案的形狀是對的。級數的前幾項是對的；積分是一個著名的值，卻配上了錯誤的上下限；因式分解停在肉眼看不出的地方。下面每一列展示模式給出的答案、它出錯的那一項，以及引擎真正計算時傳回的結果。
本頁不點名任何模型。左欄是模式比對在這些題目上普遍會給出的結果，任何一個模型一旦進步，它就過時了；右欄在每次建置時都重新對照引擎驗證，若有偏差，本頁就無法發布。
出錯的一項： 31x^9/2835
建置時已對照引擎驗證
The first four coefficients are common enough to be recalled correctly; the fifth is not. The true coefficient is 62/2835. A series that is right to four terms and wrong at the fifth looks like knowledge and is the shape of a guess.
出錯的一項： -1
建置時已對照引擎驗證
A 4x4 characteristic polynomial has enough terms for one sign to go wrong on the way to the answer, and a wrong sign produces a wrong eigenvalue that still looks like one. Here the matrix decouples into two 2x2 blocks; the engine finds all four exactly rather than approximating the quartic.
出錯的一項： Pi
建置時已對照引擎驗證
The Dirichlet integral is famous enough to be recalled, and famous enough to be recalled with the wrong limits: the value over (-inf, inf) is Pi, over (0, inf) it is Pi/2. Both facts are true; attaching the right one to the limits written down is the computation.
出錯的一項： 329218123329 declared prime
建置時已對照引擎驗證
Dividing out the 3 is easy; deciding whether the twelve-digit cofactor is prime is not something inspection can do, so a plausible answer stops there and calls it prime. The engine factors it, and it is not.
出錯的一項： 1
建置時已對照引擎驗證
The theorem is real and the shape of the problem invites it, but 10^6 is not 10^9+6. The correct residue is reached only by actually reducing, which is a few hundred squarings - trivial for an engine and impossible by inspection.
出錯的一項： Pi^4/96
建置時已對照引擎驗證
zeta(2) = Pi^2/6 is remembered by everyone; zeta(4) is remembered as 'Pi^4 over something', and the something is where a recalled answer diverges from a computed one. 90 is right; 96 and 45 both appear in nearby identities and are both wrong here.
Qurak 是一個 MCP 伺服器。連接之後，Claude、Cursor、VS Code 或任何 MCP 用戶端都會去計算而不是猜測——而且在免費方案上就開放。或者就在這裡試算一個運算式，不需帳戶。