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Five Pumpkins Weighed in Every Pair: Find the Total Weight

Jane StreetQuantitative Researcher · Onsite
MediumQuantitative ResearcherOnsiteSoftware Engineering Fundamentals

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This question evaluates combinatorial counting, algebraic reasoning, and mental-arithmetic skills by testing understanding of pairwise sums and how individual elements contribute to aggregate totals.

You have 5 pumpkins. You weigh them two at a time, in every possible pairing, and the scale readings for the ten pairs are:

$$21,\ 22,\ 23,\ 24,\ 25,\ 26,\ 27,\ 28,\ 29,\ 30 \text{ pounds.}$$

You are not told which pair of pumpkins produced which reading. What is the total weight of all five pumpkins?

This is a phone-screen mental-math question — you should be able to reason it out and get an exact number without a calculator.

Constraints & Assumptions

  • All $\binom{5}{2} = 10$ distinct pairs are weighed exactly once.
  • The scale is exact (no measurement error); treat the ten readings as given data to work with, even if their exact realizability by five real weights is not assumed up front.
  • Individual pumpkin weights are positive real numbers — they are not required to be whole pounds.
  • The ten readings are given as a multiset; the correspondence between readings and pairs is unknown.

Clarifying Questions to Ask

  • Does "any two at a time" mean all $\binom{5}{2} = 10$ possible pairs, each weighed exactly once?
  • Do we know which reading corresponds to which pair, or only the multiset of ten values?
  • Can individual pumpkins have non-integer weights, even though every pairwise reading is a whole number?
  • Is the question asking only for the total weight, or also for the individual weights?

Hints

Hint 1

Count the appearances: There are $\binom{5}{2} = 10$ pairings. If you add up all ten scale readings, how many times does each individual pumpkin get counted in that grand sum?

Hint 2

Arithmetic series: The ten readings are consecutive integers, so their sum is quick to compute: pair them up from the outside in, or use $\frac{(\text{first} + \text{last}) \times \text{count}}{2}$.

Follow-up questions

  • Can you determine the individual weights of the five pumpkins, not just the total? Which readings can you immediately identify (e.g., what must the smallest and largest readings correspond to)?
  • Generalize: $n$ objects are weighed in all $\binom{n}{2}$ pairs. Express the total weight in terms of the sum of the pairwise readings.
  • Suppose one of the ten weighings was lost and you only have nine readings. Can you still recover the total weight? What additional reasoning is needed?
  • The ten pairwise readings here are all integers. Does that force the individual pumpkin weights to be integers? Why or why not?