Problem 18 · 2016 Math Kangaroo
Stretch
Number Theory
place-value
Eight cards with the numbers 1, 2, 4, 8, 16, 32, 64, 128 are each in an unmarked envelope. Eva randomly chooses some of these eight envelopes. Ali takes the remaining ones. Both add their numbers together. They find out that Eva’s sum is 31 bigger than Ali’s sum. How many envelopes has Eva chosen?
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Answer: D — 5
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Hint 1 of 2
The eight cards are the powers of two from 1 to 128; their total is 255.
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Hint 2 of 2
If Eva's sum beats Ali's by 31 out of 255, find Eva's exact total, then read off its binary digits.
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Approach: binary split
- The cards total 1+2+…+128 = 255.
- Eva − Ali = 31 and Eva + Ali = 255, so Eva = (255+31)/2 = 143.
- 143 = 128+8+4+2+1, which is 5 cards.
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