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Source: <a href="https://www.archimedes-lab.org/monthly_puzzles_69.html" target="_blank" rel="noopener noreferrer">Previous monthly puzzles: October-November 2007</a>
Yes, the beetles will meet! Even if D0 were 8 cm (one ladybug on the pole, the other on the cannonball), the change is thus:[/FONT]
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Sec.
[/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
Band Length[/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
Bugs Gap[/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
1)[/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
D1 = 16 [cm];[/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
A1 = 12 [cm][/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
2)[/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
D2 = 24 [cm];
[/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
A2 = 15 [cm][/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
3)[/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
D3 = 32 [cm];
[/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
A3 = 17 1/3 [cm][/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
4)[/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
D4 = 40 [cm];
[/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
A4 = 19 1/6 [cm][/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
5)[/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
D5 = 48 [cm];
[/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
A5 = 20.6 [cm][/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
6)[/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
D6 = 56 [cm];
[/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
A6 = 21.7 [cm][/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
7)[/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
D7 = 64 [cm];
[/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
A7 = 22 4/7 [cm][/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
8)[/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
D8 = 72 [cm];
[/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
A8 = 23 3/35 [cm][/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
9)[/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
D9 = 80 [cm];
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A9 = 23.422... [cm][/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
10)[/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
D10 = 88 [cm];
[/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
A10 = 23.564... [cm][/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
11)[/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
D11 = 96 [cm][/FONT]; [FONT=Verdana, Arial, Helvetica, sans-serif]
A11 = 23.524... [cm][/FONT] [FONT=Verdana, Arial, Helvetica, sans-serif]
The beetles are now closing faster than the distance between them grows. After 30 seconds, the rubber band is 248 cm long and the bugs are 0.316... [cm] apart. They will meet just before the next second.[/FONT]
[FONT=Verdana, Arial, Helvetica, sans-serif]But... There is a more mathematical way to solve this puzzle without ITERATIVE calculations![/FONT]
[FONT=Verdana, Arial, Helvetica, sans-serif]Consider that each time the cannonball stretches the rubber band, the bug comes with it, which means that the FRACTION of band left to cross stays the same.[/FONT]
[FONT=Verdana, Arial, Helvetica, sans-serif]Each beetle moves specularly to its counterpart toward the central point of the rubber band. Their relative position being after each move specular to each other, to solve this puzzle we just need then to take into consideration the moves of one ladybug on a HALF section of the rubber band.[/FONT]
[FONT=Verdana, Arial, Helvetica, sans-serif]OK. After one crawl, the beetle has done 1 cm, that is 1/4 of the rubber band (fig. a.1), while the cannonball stretches the band to 8/2 = 4 cm (and the ladybug is now 2 cm along). The beetle crawls 1 cm again, which is now 1/(4 + 4) = 1/8 of the total band (fig. a.2). His next crawl will only be 1/(8 + 4) = 1/12 of the band (fig. a.3). And so on...
The fraction of band that the ladybug has crawled at 6 steps is:
1/4 + 1/8 + 1/12 + 1/16 + 1/20 + 1/24
or
(1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6)/4 [/FONT]
[FONT=Georgia, Times New Roman, Times, serif]fig. a)[/FONT]![[IMG]](proxy.php?image=http%3A%2F%2Fwww.archimedes-lab.org%2Fim_maths%2Fmon_p_114_SOLU.gif&hash=9f141c901756658d8a8e9bac569eaeb4)