Problem 16 · 2024 Math Kangaroo
Stretch
Counting & Probability
careful-countingcasework
Blind people use Braille, in which numbers and letters are shown as a code with up to 6 tactile dots (black in the picture). The numbers 0 to 9 are shown on the right. How many two-digit numbers have exactly four black dots?
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Answer: E — 23
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Hint 1 of 2
First read off how many black dots each digit 0–9 uses from the chart.
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Hint 2 of 2
A two-digit number's dots are the tens digit's dots plus the units digit's dots; count the digit pairs whose dot-counts add to 4 (remember the tens digit can't be 0).
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Approach: tabulate dots per digit, then count pairs summing to four
- From the chart, record the number of black dots for each digit 0 through 9.
- For a two-digit number, add the dots of the tens digit and the units digit; we want this total to equal 4.
- Counting all allowed digit pairs (tens digit 1–9, units digit 0–9) whose dot-counts sum to 4 gives 23 numbers.
- So 23 two-digit numbers have exactly four black dots.
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