To help celebrate the gorgeous festival of lights, a quick puzzle involving Diwali diyas.
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To decorate her home, Rukhsana has bought 20 diyas. She lays them out in a long line outside her front door. Now Rukhsana is also a mathematician. Being so, she dreams up an intriguing method to light her diyas.
First, she lights a few diyas — meaning, more than 1, but less than 10 — totally at random.
Done with that, she starts working two diyas at a time. Each move goes like this. Again at random, she picks an unlit diya and lights it. She turns to the one immediately to its right: if it’s also unlit, she lights it. If it’s lit, she extinguishes its flame.
(Exception to this pair procedure: if she picks the diya on the extreme right, she lights it.)
Question: will this process ever end? That is, will Rukhsana manage to light all 20 diyas?
Hint: Label unlit diyas with “1”, lit diyas with “0”. Consider the number that’s formed by her 20-lamp sequence of 1s and 0s. What happens on each of her pair-wise moves? There are two possibilities (well, three with the exception above). Think about them.
Scroll down for the answer.
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Answer:
The possibilities on each move:
* Both lamps Rukhsana chooses are unlit, ie, that’s 11. She lights both, turning 11 into 00.
* One is unlit and the other is lit, ie, 10. Rukhsana turns this into 01.
(And the diya on the extreme right? 1 turns into 0).
If you think of all these as numbers, in each case Rukhsana’s action decreases their value: 11 becomes 0, 10 becomes 1, and 1 becomes 0. This means she will head inexorably for an overall 0, meaning all 20 0s. Meaning, all the diyas will be lit.
So yes, Rukhsana will eventually light all 20 diyas!
A thoroughly joyous Diwali to Rukhsana, and to you and yours.
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