MHB Easier Alternatives to Tedious Tasks

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The discussion explores alternatives to tedious tasks, specifically focusing on a card flipping problem where cards start face up and are turned over based on their multiples. Cards that remain unturned are identified as those not multiples of 2, 3, 4, or 5, primarily including prime numbers greater than 5 and specific multiples. Cards turned over exactly twice are characterized as even multiples of 3 that are not multiples of 4 or 5, along with other combinations of multiples. Participants engage in calculating and identifying these card patterns, with a query about the total number of cards fitting certain criteria. The conversation emphasizes finding efficient methods to solve complex problems.
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View attachment 6362 This seems really tedious, is there a better way than powering through/?
 

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Since the cards all started with the red side up, that would be the cards that are turned over an even number of time- 0, 2, or 4. Since we turn over every card that is a multiple of 2, then 3, then 4, then 5, the cards that are never turned over are those that are that are not multiples of any of those- 7, 11, 13, 17, 19, etc. That include all prime numbers larger than 5 but also those that are multiples of 7, 11, etc. I will let you work out how many there are.

The cards that are turned over exactly twice are those that are even and a multiple of 3 but not 4 or 5- 6, 18, etc. Of course any multiple of 4, other than those that are also multiples of 3 and/or 5. Also those that are multiples of 2 and 5 but not 3 and 4, as well as those that are multiples of 3 and 5 but not 2.
 
is the answer 52?
 
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Thread 'Trigonometry problem of interest'

Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
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