23. Find the Defective Box in One Weighing You have 10 boxes, each containing exactly 10 identical-looking balls. Every normal ball weighs 10 g, but one entire box is defective: every ball in that box weighs only 9 g. You have a precise digital scale that you may use only once.How can you determine, with just that single weighing, which box holds the lighter balls? Added 3 February 2008 Show Hint Show Solution Hint: Try taking a different number of balls from each box so that the total shortage in grams directly reveals the box number. Solution: Label the boxes 1 through 10. From box 1 take 1 ball, from box 2 take 2 balls, and so on, taking 10 balls from box 10. You now have 1 + 2 + … + 10 = 55 balls.If every ball weighed 10 g, the scale would read 55 × 10 = 550 g. Place all 55 balls on the scale and note the actual reading.Because the balls from the defective box weigh 9 g instead of 10 g, each ball taken from that box makes the total weight 1 g lighter than expected. The total weight will therefore be:550 g − (box number) gFor example, if the scale shows 546 g, the shortage is 4 g, so box 4 is defective. Whatever number of grams is missing from 550 immediately identifies the faulty box.
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