Daily Math Puzzle: 2026-06-29
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2026-06-29
Deep within the Whisperwood Labyrinth, the rogue Elara stumbles upon a chamber guarded by a mischievous sprite. To pass, Elara must solve the 'Gauntlet of Flickering Flames'. A row of 100 magical lanterns, numbered 1 to 100, hang from the ceiling, all initially extinguished. The sprite explains the challenge: 'You must walk through this chamber 100 times. On your first pass, you will touch and toggle every single lantern. On your second pass, you will touch and toggle every second lantern. On your third pass, you will touch and toggle every third lantern, and so on. This continues until your 100th pass, where you will touch and toggle only the 100th lantern. Only when you tell me how many lanterns will be lit after all 100 passes will the path to freedom open!'
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Solution
10 — A lantern will be lit at the end if it has been toggled an odd number of times. A lantern numbered 'N' is toggled during pass 'K' if 'K' is a factor of 'N'. Therefore, a lantern will be lit if it has an odd number of factors.
The only numbers with an odd number of factors are perfect squares.
We need to count the perfect squares between 1 and 100 (inclusive):
1^2 = 1
2^2 = 4
3^2 = 9
4^2 = 16
5^2 = 25
6^2 = 36
7^2 = 49
8^2 = 64
9^2 = 81
10^2 = 100
There are exactly 10 such numbers. So, 10 lanterns will be lit.
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