Double One, Halve the Other

Question

Compute 16 times 35 in your head in two seconds. Then state when the double–half move pays off, and when it is the wrong tool.

Accepted Answer

The move: halve the even factor, double the one ending in 5, doubling a 5-ender mints a trailing zero, and trailing zeros are free: 16 times 35 = 8 times 70 = mathbf560. Iterate while it keeps paying: 25 times 36 = 50 times 18 = 100 times 9 = 900, three slides turn a two-digit product into a one-digit one. Calibration: 45 times 14 = 90 times 7 = 630; 35 times 12 = 70 times 6 = 420; 4.5 times 16 = 9 times 8 = 72 (decimals ride along). When it pays: one factor ends in 5 or 50, or one factor is a power of 2 (you can halve repeatedly without ever hitting a fraction: 16 = 2^4 buys four halvings). When it is wrong: both factors odd, halving creates fractions, so anchor-and-adjust (17 times 13 = 17 times 13 via 170 + 51) beats it. The screening test is one glance: can I manufacture a trailing zero? The underlying principle is broader than 2s: products are invariant under ab = fracak times kb for any k. The same lens makes times 5 instant (times 10 then halve: 5 times 86 = frac8602 = 430) and powers the ÷25 and ÷125 family. On Flow Traders and Optiver screens, the candidates who finish early are the ones who spend their first half-second re-shaping the product instead of attacking the digi

Double One, Halve the Other

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