Para-jumble (Hardy–Ramanujan anecdote): Arrange the sentences to recreate the famous taxi-cab story, moving from Hardy’s awkward greeting to Ramanujan’s punchline and its mathematical significance. S1 = "There is a touching story of Professor Hardy visiting Ramanujan as he lay desperately ill in hospital at Putney." S6 = "It is the lowest number that can be expressed in two different ways as the sum of two cubes." Between S1 and S6, order the fragments into a coherent narrative: P = "No Hardy, that is not a dull number in the very least." Q = "Hardy, who was a very shy man, could not find the words for his distress." R = "It was 1729." S = "The best he could do, as he got to the bedside, was "I say, Ramanujan, I thought the number of the taxi I came down in was a very dull number."" Choose the correct sequence of P–Q–R–S that completes the paragraph.

Given data

• S1 sets the scene: Hardy visits Ramanujan in hospital.
• Q: Hardy is very shy and struggles to speak.
• S: He finally utters an awkward line about the taxi number being dull.
• R: The number mentioned is 1729.
• P: Ramanujan replies that it is not dull at all, anticipating S6.
• S6 explains why 1729 is remarkable: two different representations as sums of two cubes.

Concept / Approach (narrative causality)
The paragraph follows a natural dialogue flow: character trait → hesitant remark → reveal of the number → witty corrective response → mathematical explanation. Starting with Q motivates why S is awkward; R supplies the concrete value that P can then reframe, leading smoothly into S6.

Step-by-step construction
S1 introduces the hospital visit.
Step 1: Set the speaker's difficulty (Q)
"Hardy, who was a very shy man, could not find the words for his distress."
Step 2: Provide the awkward remark (S)
"The best he could do ... was 'I say, Ramanujan, I thought the number of the taxi I came down in was a very dull number.'"
Step 3: Reveal the number (R)
"It was 1729."
Step 4: Give Ramanujan's famous reply (P)
"No Hardy, that is not a dull number in the very least."
Close with S6 (reason)
"It is the lowest number that can be expressed in two different ways as the sum of two cubes."

Verification (why 1729 is special)
1729 = 1³ + 12³ = 1 + 1728 = 17291729 = 9³ + 10³ = 729 + 1000 = 1729Thus, it is the smallest number expressible as the sum of two positive cubes in two distinct ways.

Why the other orders fail
PRSQ: Starts with Ramanujan's rebuttal before Hardy even mentions the number; dialogue logic breaks.QSPR: Puts the witty reply after Q and S but delays the identity of the number (R) too long; the rebuttal lacks its immediate numerical referent.SQRP: Begins with the remark before establishing Hardy's shyness (Q), and ends with P but leaves no space for S6's explanation to connect.

Common pitfalls

• Omitting the numeric reveal (R) before the corrective statement (P), which undermines coherence.
• Ignoring that S6 is an explanatory tail that must follow the dialogue exchange.
• Letting the narrative jump back and forth in time instead of following the hospital-visit chronology.

Final Answer
QSRP