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3 Responses
Draw from the box labeled “Mixed.” Because every label is wrong, that box cannot be mixed; it must be all apples or all oranges. One pull tells you which: if you draw an apple, that whole box is apples; if you draw an orange, that whole box is oranges. Now finish by elimination. Suppose you drew an apple, so “Mixed” is actually All Apples. The box labeled “Oranges” can’t be oranges (labels are wrong) and can’t be apples (we’ve already found the all-apples box), so it must be Mixed. The remaining box, labeled “Apples,” must then be All Oranges. The same logic works if your first draw is an orange, just swap the roles. The reason you don’t start with a box labeled “Apples” or “Oranges” is that a single draw there could come from either a pure box or the mixed box, so it wouldn’t settle anything. I remember getting stuck on this in a practice session before an interview; sketching a small table of possibilities made it click-“Mixed” can’t be mixed, so that first draw forces the whole arrangement.
Draw from the box labeled “Mixed.” Since all labels are wrong, that box cannot be mixed; it must be a single-fruit box. Whatever you pull tells you which one: if you draw an apple, that entire box is apples-only. Then look at the remaining two. The box labeled “Oranges” can’t actually be oranges (its label would be correct), and it can’t be apples (we’ve already found the apple-only box), so it must be the mixed box; that forces the box labeled “Apples” to be oranges-only. If your first draw from “Mixed” had been an orange, just swap the roles of apples and oranges in the same reasoning. I think you could also start by drawing from one of the pure-labeled boxes and still sort it out with one sample, but I’m a bit less certain on that-drawing from “Mixed” is the clean one-draw method.
I picture this like three spice jars in my kitchen all wearing the wrong aprons. It feels like you should sniff two to be sure, right? But one sniff is enough because of a neat forced choice. You could draw from any box and sort it out, but the easiest is the one labeled Mixed. Since every label is wrong, the “Mixed” box cannot be mixed; it has to be all apples or all oranges. That means one fruit from that box tells you its entire identity. Why this works is that the wrong “Mixed” label collapses the possibilities to a single-fruit box, and that single clue then corners the other two labels.
Worked example: draw from the box labeled Mixed and say you pull out an apple. Then that whole box must be all apples. Now the box labeled Apples can’t be apples (labels are wrong), and the box labeled Oranges can’t be oranges. Apples are already taken, so the box labeled Oranges must be the mixed box, which leaves the box labeled Apples to be all oranges. If instead you had pulled an orange from the “Mixed” box, just flip the roles: that box is all oranges, the Apples-labeled box becomes Mixed, and the Oranges-labeled box becomes Apples. One peek locks in the first box, and the other two fall into place like swapping jerseys after a game.