Difficulty: Tricky
Ahmad’s computer scientist mother gives him a binary stopwatch that counts minutes using only 1’s and 0’s – just like a computer! The binary stopwatch has 6 placeholders where the 1’s and 0’s appear. When the binary stopwatch is reset, all the placeholders are zeros.
Ahmad finds a regular stopwatch, resets both stopwatches, and starts them both at the same time. He records when the binary clock’s zeros turn into ones, like so:
If Ahmad’s binary stopwatch continues this pattern, how many minutes will the regular stopwatch show when the binary stopwatch reads 100000?
Bonus question:
This stopwatch can also show numbers with more than one 1. For example:
What’s the highest number that this stopwatch can show?
Scroll down or click for a hint, or the answer!
Brainteaser hint
Check out the pattern that’s forming: 1, 2, 4, 8. What numbers come next in this sequence?
For the bonus question, check out the binary stopwatch at a few other times:
Can you use Ahmad’s pattern and simple patterns of addition to convert between the binary stopwatch and the regular one?
Brainteaser answer
Ahmad’s regular stopwatch will read 32 minutes when 100000 appears on the binary stopwatch.
Bonus question: the binary stopwatch can display up to 63 minutes.
Ahmad notices new placeholders changing at 1, 2, 4, and 8 minutes. Each of those numbers is 2 times the previous number. So, we would expect:
There’s not a lot of information for the bonus question unless you read the hint, so if you’re having a hard time, go take a look!
We’re told that 001011 = 11, which doesn’t seem like enough information. But we know some other things too. For example:
001000 = 8
000010 = 2
000001 = 1
If we add up the left side of these 3 equations, we get 001011, and adding up the right side gets us 11, the answer we were looking for!
You can check this idea with the other examples in the hint to make sure it always works.
Based on this information, the biggest possible number would be 111111.
100000 = 32
010000 = 16
001000 = 8
000100 = 4
000010 = 2
000001 = 1
Adding them all up gives us the answer:
111111 = 32+16+8+4+2+1 = 63
Congrats! You cracked the computer clock. For more about binary, check out our post, Binary for beginners here.
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