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Rotation Bracket
divider mounting brackets to the TELEVISION -- a superb preference
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Maths Challenge?
We place some positive whole numbers around a circle.The sum of each pair of neighbouring numbers is written in a box drawn in the space between them(in this case [] signifies box) We call this arrangement a sum cirlce.Here is an example using number 1,2,3,4,5,6 1[6]5[7]2[8]6[9]3[7]4[5]
Like a bracket, a sum circle is not changed by rotations and reflections.For example these two sum circle are the same as the one above 3[7]4[5]1[6]5[7]2[8]6[9] and 5[6]1[5]4[7]3[9]6[8]2[7]
If all the boxes contain prime numbers it is called a sum circle
a)Find all the dif ways of using 1,2,3,4,5,6,7,8,9,10 in a prime sum circle with 7 dif prime no's in the boxes. Xplain why ther are no more ways.
b)Show that when 1,2,3,4,5,6,7,8,9,10 are aranged into a prime sum circle, the boxes contain at least 3 prime no's.(convincin argument)
c)Find all dif prime sum circles using1,2,3,4,5,6,7,8,9,10 for which ther r exactly 3 dif prime no's in boxes.Xplain y ther r no mor circles
qa
7 primes ≥ 3 : 3, 5, 7, 11, 13, 17, 19
sum of sum circle
= 2(1+2+3+4+5+6+7+8+9+10)
= 2 * 10(1+10)/2
= 110
sum of the repeated primes
= 110 - (3 + 5 + 7 + 11 + 13 + 17 + 19)
= 110 - 75
= 35
35 = x + y + z
x,y,z are 3 of these : {5,7,11,13,17}
they can be repeated too, like x = y.
(11,11,13) or
(5,13,17) or
(7,11,17) or
(11,17,17).
3 = 1+2 ONLY ; use (1,2) and
19 = 10+9 ONLY. (9,10), because that's the only choice you have.
5 = 1+4 = 2+3 ; use (2,3)
17 = 10+7 = 9+8 ; use (8,9)
7 = 1+6 = 2+5 = 3+4 ; use (3,4)
13 = 3+10 = 4+9 = 5+8 = 6+7 ; use (6,7) and (5,8)
11 = 1+10 = 2+9 = 3+8 = 4+7 = 5+6 ; use (1,10) and (4,7) and (5,6), as for (11,11,13) repeated combination.
(11,11,13) set :
2 [3] 1 [11] 10 [19] 9 [17] 8 [13] 5 [11] 6 [13] 7 [11] 4 [7] 3 [5] 2
now look at 1 [11] 10 and 6 [13] 7.
11 - 4 = 7 (still a prime) and
13 + 4 = 17 (also maintain the primey).
so we could do some "twisting" here.
2 [3] 1 [7] 6 [11] 5 [13] 8 [17] 9 [19] 10 [17] 7 [11] 4 [7] 3 [5] 2
(11,11,13) or
(5,13,17) or
(7,11,17) or
(11,17,17).
i wrote there were 4 possible sets for the extra 3 repeated primes, right? but the (11,17,17) is impossible, because there are only 2 ways about getting 17, not 3.
and for (5,13,17), both 5 and 17 have the only 2 combination each, so you have 2 short chains of 3-2-1-4 (or 4-1-2-3) and 7-10-9-8 (or 8-9-10-7). the 2 numbers to be used are 5 and 6. you can connect in 2 ways : "chain - chain - 5 - 6", or "chain - 5 - chain - 6". by now we must have noticed the even-odd-oven-odd pattern, so for "chain - chain combination", it's either 3-2-1-4-7-10-9-8 (and it's mirror) or 4-1-2-3-8-9-10-7 (and it's mirror). inserting 5 and 6 in either would not work (because 3+6=9 and 4+5=9).
for "chain - 5 - chain - 6" combination, 5 must be connected to the even ends (4 and 
and 6 to odd ends (7 and 3). 4+5 and 6+3 made it not work.
for (7,11,17) set, we get
2 [3] 1 [5] 4 [7] 3 [11] 8 [17] 9 [19] 10 [17] 7 [13] 6 [11] 5 [7] 2
and
2 [3] 1 [7] 6 [11] 5 [13] 8 [17] 9 [19] 10 [17] 7 [11] 4 [7] 3 [5] 2
qb
if you arrange like this :
4 [11] 7 [13] 6 [11] 5 [13] 8 [11] 3 [13] 10 [11] 1 [3] 2 [11] 9 [13] 4
you get only 3 primes ; namely 11 (5 of them), 13 (there's 4) and 3 (the only one, summing the 2 smallest number).
how could they came about? to contain the least number of distinc primes, you need to "stabilize" between 2 prime sum. like if we arrange the evens in ascending order (2-4-6-8-10), we need to balance the ascending-evens (e --> E) out with descending odds (O --> o), like (9-7-5-3-1).
we'll get 2 alternating, "stabilized" prime sum, 11 and 13, though we could also have 9 and 11, but 9 is not a prime. 11 is there because we have a lump total of 110, and divided into 10 boxes, we get 11. 9 and 13 are 11's 2 odd neighbours, but 13 is the one that's prime.
but our long chain, 2-9-4-7-6-5-8-3-10-1, had to end with either the biggest or the smallest sum. if it was 9 and 11, the other prime would be 19 (but 9 is not, so disregard them). we get 11,13 and 3 least triple primes.
qc
4 [11] 7 [13] 6 [11] 5 [13] 8 [11] 3 [13] 10 [11] 1 [3] 2 [11] 9 [13] 4 --> (as previous answer)
the rules are these below :
110 = ap + bq + cr
p,q,r are 3 distinct primes in {3,5,7,11,13,17,19}
a + b + c = 10
for previous answer,
a = 5, p = 11, b = 1, q = 3, c = 4, r = 13.
for (a,b,c) = (5,2,3) and (p,q,r) = (11,17,7), we get
4 [11] 7 [17] 10 [11] 1 [7] 6 [11] 5 [7] 2 [11] 9 [17] 8 [11] 3 [7] 4
you could try with other applicable a, b, c, p, q and r, as long as the rules is preserved.
110 = ap + bq + cr
0 < a, b, c < 6
a + b + c = 10
3 (1) = 3
5 (2) = 10
7 (3) = 21
11 (5) = 55
13 (4) = 52
17 (2) = 34
19 (1) = 19
4 [11] 7 [13] 6 [11] 5 [13] 8 [11] 3 [13] 10 [11] 1 [3] 2 [11] 9 [13] 4
and
4 [11] 7 [17] 10 [11] 1 [7] 6 [11] 5 [7] 2 [11] 9 [17] 8 [11] 3 [7] 4
are the only 2 answers.
4#3B "anti-rotation-bracket-less" demo
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