Esempi svolti
Tre domande pubbliche in stile Scientific Thinking, ciascuna rivista per intero: il passaggio che la decide, una ragione per ogni opzione e il metodo da ricordare per la prossima domanda dello stesso tipo. Le domande e il ragionamento sono in inglese, come all’HUMAT.
Esempio 1
Simboli al posto delle variabili.
Una domanda di pensiero matematico dalla calibrazione gratuita. Tre equazioni, un valore da trovare e quattro modi per indicarne uno sbagliato.
Mathematical thinking · symbols in place of variables
The symbols ◆, ● and ▲ each stand for an integer. They satisfy:
◆ + ● + ▲ = 5
2◆ − ● + ▲ = −3
◆ − ● + ▲ = −1
What is the value of ●?
Il passaggio decisivo
Equations 1 and 3 differ only in ●. Subtract one from the other and ◆ and ▲ cancel: 2● = 6, so ● = 3. The full solution is ◆ = −2, ● = 3, ▲ = 4.
Il perché di ogni opzione
- A−2EsclusaThis is the value of ◆. Solving the system correctly and then reporting the wrong symbol is the easiest way to lose this mark.
- B4EsclusaThis is the value of ▲: the same trap, a correct solution with the wrong symbol reported.
- C−3EsclusaA sign slip. Equation 3 minus equation 1 gives −2● = −6; dropping one minus sign leaves ● = −3, and equation 1 no longer holds.
- D2EsclusaThe size of ◆ with its sign lost. No value of ● other than 3 satisfies equations 1 and 3 together.
- E3Risposta correttaEquation 1 minus equation 3 cancels ◆ and ▲ and leaves 2● = 6.
Il metodo da ricordare. Before solving a system, look for two equations that differ in a single term. Subtracting them isolates that term in one step.
Esempio 2
Raccogliere elementi da insiemi diversi.
Una domanda di pensiero procedurale. Quattro regole, cinque insiemi possibili e una regola che decide il resto.
Procedural thinking · collecting elements from different sets
Five samples, A to E, may be included in a set. The set must satisfy four rules:
- Exactly one of A and B is included.
- D is included.
- If C is included, D is not.
- Exactly one of C and E is included.
Which set satisfies every rule?
Il passaggio decisivo
D is forced in. C would force D out, so C is out, and then E must be in. Only A, D and E keeps D and E with exactly one of A and B.
Il perché di ogni opzione
- AA, C and DEsclusaC appears together with D, which the third rule forbids.
- BA, D and ERisposta correttaD and E are in, C is out, and exactly one of A and B is in: every rule holds.
- CB and DEsclusaNeither C nor E is included, but exactly one of them must be.
- DA, B, D and EEsclusaA and B are both included; exactly one of them is allowed.
- EB, C, D and EEsclusaC appears with D, and C and E are both in: two rules fail.
Il metodo da ricordare. Start from the rule that fixes something outright, follow its consequences, and only then read the options.
Esempio 3
Seguire un protocollo.
Una domanda di pensiero procedurale. Una procedura in tre passaggi e una proprietà che esclude due opzioni prima di qualsiasi calcolo.
Procedural thinking · following a protocol
A sorting protocol gives each sample a code. Start with the sample number n.
- If n is even, divide it by 2. If n is odd, add 5.
- If the result is greater than 10, subtract 4. Otherwise, multiply it by 3.
- The code is the remainder when the result is divided by 4.
Which sample number receives code 3?
Il passaggio decisivo
Work backwards. An odd n becomes even at step 1, and both branches of step 2 keep it even, so its code is 0 or 2: 9 and 11 are out at once. Of 6, 10 and 14, step 2 gives 9, 15 and 21; only 15 leaves remainder 3.
Il perché di ogni opzione
- A6Esclusa6 → 3 → 9, and 9 leaves remainder 1: code 1.
- B9Esclusa9 is odd, so step 1 gives 14. 14 → 10, which leaves remainder 2: code 2.
- C10Risposta corretta10 → 5 → 15, and 15 = 3 × 4 + 3: code 3.
- D11Esclusa11 → 16 → 12: code 0. Like every odd number, it can only reach code 0 or 2.
- E14Esclusa14 → 7 → 21, and 21 leaves remainder 1: code 1. It takes the same route as 10 and lands on the wrong remainder.
Il metodo da ricordare. When a procedure ends in a property, ask which inputs can produce that property at all. Here parity removes two options before any arithmetic.
Prova altre otto domande, a tempo.
La calibrazione gratuita è fatta di otto domande di Scientific Thinking a tempo, al ritmo dell’esame, ciascuna con la soluzione commentata. 18 minuti, senza account.