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Maths challenge

AuthorMessage
dewtell*

Posts: 26

new Posted: 12:55PM Feb 5, 2015

Ok, lets try for an easier one this time. The number 21 isn't a prime, but it could be changed into one by either changing the first digit (to 11) or the second digit (to 23). What is the smallest positive integer that is neither a prime itself, nor can it be changed into a prime by changing one digit?
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dewtell*

Posts: 26

new Posted: 04:49PM Feb 5, 2015

And JQ, to answer the question you intended to ask, the discriminant of that quadratic is 16 cos^2 θ - 24 sin θ, and the denominator is 6 sin θ. To have distinct real roots, we need the discriminant > 0 and the denominator non-zero. Rewriting the discriminant, and factoring out 8, we have -2 sin^2 θ - 3 sin θ +2 > 0, or 2 sin^2 θ + 3 sin θ - 2 < 0. Factoring the LHS, we get: (2 sin θ -1)(sin θ + 2) < 0. The second factor will always be positive, so this reduces to 2 sin θ - 1 < 0, or sin θ < 1/2. Combine with the requirement that sin θ =/= 0, and we should get solutions where (-7π/6 + 2πk < θ < π/6 + 2πk) and (θ not an integral multiple of π).
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JQPublic*

Posts: 1887

new Posted: 08:45AM Feb 6, 2015

According to the prime number list (hope that doesn't count as cheating ...), there is at least one prime for every 10-number interval below 200.

200 seems to be the answer. Since the last digit is zero, obviously we can't get any primes by changing the first two digits. 201 (3), 202 (2), 203 (7), 204 (2), 205 (5), 206 (2), 207 (3), 208 (2), 209 (11) are all composite.

P.S. The 'trap' intended by the book is sin θ =/= 0, since it's easy to forget that.


---This message was edited on 08:46AM Feb 6, 2015---

'An idea, like a ghost, must be spoken to a little before it will explain itself.' - Charles Dickens
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dewtell*

Posts: 26

new Posted: 11:09AM Feb 6, 2015

200 is correct. As you noted, the key is to see that you are looking for the first prime-free decade, which you can get to by noting that the last digit of the number must be 0 (because if it wasn't, then you could get a smaller number from whatever the answer was by changing the last digit to 0), and then noting that if the last digit is 0, the only digit that is a candidate for changing is that last digit.

I was able to solve it in my head without a prime list by noting that the key numbers in any decade are those ending in 1, 3, 7, and 9, because the others will be non-prime for any decade but the first. 3 will divide at most 2 of those (either 1 and 7 or 3 and 9), because any number divisible by 3 must have a digit sum divisible by 3, and 7 will divide at most 1 of those, so we need to have at least one of the numbers with a smallest prime factor of 11 or greater. So the first possible decade is the one including 121. 123 and 129 are divisible by 3, but 127 isn't divisible by 7, so it won't work. The next possible decade is the 140s, where we add 22 to 121 to get 143. 141 and 147 are divisible by 3, but 149 isn't divisible by 7, so it won't work. The 160s has 169 divisible by 13, but otherwise is completely unpromising, since both 3 and 11 hit 165 instead of the four numbers we care about. 13 is a non-factor for the 180s (182), 11 hits 187, and 3 gets 183 and 189, but 181 isn't a multiple of 7. 13 won't be any help in the 190s (195), so we skip to the 200s, where 11 gets 209, 3 gets 201 and 207, and 203 *is* divisible by 7. Bingo!

Your turn to pose.
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JQPublic*

Posts: 1887

new Posted: 10:42AM Feb 7, 2015

Hmm. I gave this some thought and eventually decided to copy from a book again.

CDE is a triangle where CE = DE. B is a point on DE and F is a point on CE. BF is produced to A such that AB = AC. Angle BAC and angle DEC are equal. Prove that AE and CD are parallel.


'An idea, like a ghost, must be spoken to a little before it will explain itself.' - Charles Dickens
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dewtell*

Posts: 26

new Posted: 05:19PM Feb 8, 2015

I tried to see if I could create an example of your triangle problem using right angles (to make the problem a little easier), and wound up finding a counterexample. On a standard cartesian grid, let C(4,0), D(0,4), E(0,0), and B(0,2) be points for this problem. Then there are two possible locations for A satisfying the constraints of the problem: A(3,3), implying F(-6,0), and A(1,-1), implying F(2/3,0). With the first location, length(AB) = length(AC) = sqrt(10), and angle BAC is a right angle congruent to DEC, but AE and CD are perpendicular, not parallel. Only the second location for A results in them being parallel.

It might be the case that if F is constrained to lie between E and C, then the lines must be parallel, but it's not true in general. For a given set of locations for C, D, E, and B, there will be two possible locations for A that have the angles match: both located on the perpendicular bisector of BC, one on each side of BC. At most one of those will yield parallel lines for AE and CD.
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JQPublic*

Posts: 1887

new Posted: 07:55AM Feb 9, 2015

dewtell wrote:
I tried to see if I could create an example of your triangle problem using right angles (to make the problem a little easier), and wound up finding a counterexample. On a standard cartesian grid, let C(4,0), D(0,4), E(0,0), and B(0,2) be points for this problem. Then there are two possible locations for A satisfying the constraints of the problem: A(3,3), implying F(-6,0), and A(1,-1), implying F(2/3,0). With the first location, length(AB) = length(AC) = sqrt(10), and angle BAC is a right angle congruent to DEC, but AE and CD are perpendicular, not parallel. Only the second location for A results in them being parallel.

It might be the case that if F is constrained to lie between E and C, then the lines must be parallel, but it's not true in general. For a given set of locations for C, D, E, and B, there will be two possible locations for A that have the angles match: both located on the perpendicular bisector of BC, one on each side of BC. At most one of those will yield parallel lines for AE and CD.


Sorry, my bad again. the original problem was a diagram and I translated it into words. Clearly, my translation was quite muddy (no oxymoron intended there). When I said BF was produced to A, I thought it would be understood that BF is extended in the direction from B to F, but I guess this isn't the case. When I said F is a point on CE, I meant the line segment CE, not the line CE, but again I guess I wasn't clear there. Hope this clears things up.


'An idea, like a ghost, must be spoken to a little before it will explain itself.' - Charles Dickens
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