The puzzle
Several balls, all identical to look at, all different in weight. You have a balance scale and a limited number of weighings. Put them in order, lightest to heaviest.
This is sorting by comparison — the same problem that sits underneath every sorting algorithm in computer science — turned into a deduction puzzle. What makes it interesting is that the weighing budget is tight but sufficient, so guessing will not work and neither will comparing everything.
Why the budget is what it is
Sorting n items by pairwise comparison needs at least
⌈log₂(n!)⌉ comparisons in the worst case — an information-theoretic floor,
not a matter of cleverness. Each weighing gives you one bit, and you need enough bits to
distinguish n! possible orderings.
For 5 balls that is 7 weighings; for 6 it is 10; for 7 it is 13. The budgets here are set at or just above those floors, which means a perfect score is achievable but leaves almost no room for a wasted weighing.
Strategy
- Use binary insertion, not bubbling. Sort three balls first, then place each remaining ball by comparing it against the middle of what you have already ordered. Comparing against the ends wastes the bit.
- Never repeat a comparison you can deduce. If A < B and B < C, you know A < C. Spending a weighing to confirm it is a wasted weighing you cannot afford.
- Think before the first weighing. Decide which comparison splits the possibilities most evenly. A weighing that rules out half the orderings is worth two that rule out a quarter each.
- Write it down. Holding a partial order of six items in working memory is genuinely hard, and it is not what the task is testing.
Why assessments like this task
Variants of this appear in cognitive hiring assessments because it separates two things that usually travel together: having a method and executing it. Someone comparing pairs at random can still finish, badly. Someone with a strategy finishes with weighings to spare.
It also rewards planning before acting, which is unusual — most timed tasks reward starting immediately. Here the first ten seconds of thought pay for themselves.
Frequently asked
Q. Is a perfect score always possible?
A. Yes, the budgets are set at or above the theoretical minimum. But at the higher
levels there is essentially no slack, so one wasted weighing costs you the perfect score.
Q. Do the colours or positions mean anything?
A. No. Colour assignment and display order are randomised every round precisely so that
nothing carries over between attempts.
Q. Is this a memory test or a logic test?
A. Mostly logic, but memory load is real at 6 and 7 balls. Writing down the partial
order removes the memory component and leaves the reasoning, which is the interesting part.
For a planning puzzle with no hidden information at all, try the tower puzzle.