Problems & Puzzles: Puzzles

Puzzle 130.  The Hexagon of the 19 numbers

An already solved puzzle is to allocate the first 19 natural numbers (1 -19) in the hexagon shown below, such that every one of the 12 sums of 3 circled numbers in line are equal.

Questions:

Get the least solution using only distinct prime numbers not necessarily consecutive (a big clap if they are consecutive, of course)

_________
Note1:
Each of these 12 sums is over the side of one of the 6 triangles.

Note2: As a matter of fact I have seen two distinct solutions using the first 19 numbers. Are there more? Reference will be given one week later.

Solution

Well, or this puzzle seemed uninteresting to our readers or resulted a kind of hard. Time ago one puzzle lasted one complete week without being solved, at least partially.

The two solutions that I saw using natural numbers 1-19 can be found as 'trivia' at the margin of page 191 of the book "Exploring Number Theory with Microcoputers" by Donald D. Spencer.

One solution has 2 as the central number and the twelve sums are 22. The second solution has 6 as the central and the twelve sums are 23:

My friend Jaime Ayala found a third solution and made the observation that once we find one solution we may obtain its "complementary solution" just substituting each number by its complement to 20.

But no one has found at least one solution using only pure distinct primes. So the puzzle is still open!

***

Enoch Haga produced (24/3/01) the first prime-solution:

11299 6353 20341
14081 10133 1091 421
12613 8819 16561 4201 17231
5527 1579 3671 3001
19853 379 17761

Common sum = 37993 (prime!)

Later he reported a smaller solution (sum = 36847). His code (Ubasic) and method (MonteCarlo approach) should produce more solutions. I will wait a few days for his apparent-smallest solution.

Other approaches? How far are the Enoch solutions from the smallest solution?

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Jean Charles Meyrignac soon (27/3/01) solved - and more than that! - this puzzle:

a) solutions with the smallest sum: Sum=95

7 71 17
41 83 73 67
47 43 5 79 11
19 61 37 31
29 13 53

b) solutions with the smallest highest term (79): Sum=131

5 73 53
59 79 31 7
67 17 47 13 71
3 23 43 19
61 29 41

c) hexagons that have the smallest consecutive primes: Sum=223

89 31 103
37 61 47 41
97 53 73 71 79
59 83 107 101
67 113 43

Note: This is not truth, as was advised by Sudipta Das (see below his complete report) because the prime 109 is not in the list pf these 19 primes. Jean Charles consulted, admitted a bug in his code. Later Sudipta added "I have finished checking all sets of 19 consecutive primes below 10000 . There is no solution "

d) Other sums by him are: 95, 101, 103, 107, 113, 121, 125, 127, 131, 133, 137, 139, 143, 149, 151, 155, 157, 161, 163, 167, 169, 173, 179, 181, 187, ...

His method: "I used an interesting trick to reduce computations, since you can notice that the center of the hexagon is included in 6 sums.

a b c
d e f g
h i j k l
m n o p
q r s

j+a+e = j+f+c = j+h+i = j+k+l = j+n+q = j+o+s

so, we have: a+e = c+f = h+i = k+l = n+q = o+s

meaning that this sum can be decomposed in 6 different ways. So I added a small table to check how much decomposition exist for one number. If the decompositions are below 6, then this sum can't be a+e, etc... "

***

Jean Brette shows (28/3/2001) how to get a prime solution such that all the 19 primes are in A.P.

"...You can find some other solutions as follow (but not so small !). Suppose you have a "good" hexagon with numbers 1-19 and constant sum S (for example, this one, with sum S = 25)

5 18 2
12 16 19 17
8 13 4 15 6
3 7 11 9
14 1 10

Subtracting 1 to each number, you get a new "good" hexagon with numbers 0-18, and sum (S-3). Now multiply all the numbers by a constant d, and add to each an other constant a. This give you a "good" hexagon with numbers:

a, a+ d, a+ 2d, ... , a+18 d., and sum 3a+d(S-3).

If these 19 numbers are prime , you win. As you know, such arithmetic progressions with primes exist, and you can find some of them in R.K Guy : Unsolved problems in number theory, 2nd edition, p. 16. Cordially Jean"

Later he sent other non-prime solutions for a sum of: 25, 27, 28 & 31. Are this all?...

***

This summarizes the Brette's results for the solutions with numbers 1-19,  sent the 10/4/01:

"Here is the table of results ( nb) is the number of distinct solutions

C = 2 S= 22, 25, 29 and no other
C = 4 S= 25 (2), 26, 28, 29, 31 and no other
C = 6 S= 23 (your) , 24, 28, 29, 33 and no other
C = 8 S= 24, 25, 26 (2), 27 (4), 28, 31 and no other
C = 10 S= 28, 29 (2) + symmetry and no other"

 C=2 C=4 C=6 C=8 C=10 S = 22 1 18 3 13 19 17 14 8 12 2 15 5 10 16 9 6 4 7 11 S = 25 10 1 14 9 11 7 3 6 15 4 13 8 17 19 16 12 2 18 5 S = 23 1 19 3 18 16 14 12 4 13 6 9 8 17 15 7 5 2 11 10 S=24 5 7 12 18 11 4 9 1 15 8 13 3 17 10 14 19 6 16 2 S = 28 5 6 17 9 13 1 8 14 4 10 15 3 12 16 11 18 2 19 7 S = 25 8 1 16 13 15 7 3 4 19 2 17 6 10 12 18 14 11 9 5 S = 25 8 1 16 14 13 5 7 3 18 4 19 2 12 11 15 17 10 9 6 S = 24 3 8 13 10 15 5 9 11 7 6 16 2 12 17 14 18 1 19 4 S=25 2 9 14 17 15 3 4 6 11 8 10 7 18 16 12 13 1 19 5 S =29 3 19 7 15 16 12 4 11 8 10 1 18 5 6 17 9 13 14 2 S = 29 8 7 14 6 19 13 4 15 12 2 16 11 5 18 10 1 9 3 17 S = 26 9 1 16 7 13 6 2 10 12 4 14  11 17 19 15 5 18 3 S = 27 7 1 19 16 14 2 5 4 17 6 18 3 13 11 12 15 10 8 9 S=26 14 1 11 10 4 7 9 2 16 8 12 6 19 13 15 17 5 18 3 S = 29 12 3 14 16 7 5 9 1 18 10 13 6 11 2 15 19 17 8 4 S = 28 5 17 6 12 19 18 7 11 13 4 9 15 1 8 14 3 16 2 10 S = 27 7 1 19 9 14 2 5 11 10 6 18 3 12 17 13 16 4 15 8 S=26 14 1 11 9 4 7 10 3 15 8 13 5 17 12 16 19 6 18 2 S = 29 11 2 16 12 14 9 5 6 19 4 17 8 13 15 7 3 10 1 18 S=33 19 1 13 4 8 14 2 10 17 6 9 18 7 11 15 3 16 5 12 S=27 9 1 17 11 10 2 4 7 12 8 13 6 15 14 16 18 5 19 3 S = 31 11 7 13 1 16 14 6 19 8 4 15 12 3 18 10 2 9 5 17 S=27 9 3 15 16 10 4 11 2 17 8 18 1 19 13 5 12 6 7 14 S=27 6 3 18 19 13 1 4 2 17 8 14 5 16 10 12 15 9 11 7 S =27 16 9 2 5 3 17 10 6 13 8 4 15 14 12 18 11 7 19 1 S =28 4 7 17 10 16 3 9 14 6 8 18 2 13 19 5 11 1 12 15 S =31 4 18 9 15 19 14 5 12 11 8 6 17 3 7 10 1 16 2 13

***

Pierre Audin found (13/4/01) a prime & minimal solution distinct than the published from Meyrignac:

7 71 17
41 83 73 19
47 43 5 31 59
37 79 67 13
11 61 23

BTW, I should say that Meyrignac sent 48 distinct solutions of these (prime & minimal S=95).

***

Another massive attack to this puzzle came from Sudipta Das (25/11/01):

Let the hexagon be :

a1         a2          a3

a12       a13       a14        a4

a11        a16       a15        a17      a5

a10      a18        a19        a6

a9         a8         a7

(To avoid solutions due to reflection or rotation , I have placed the lowest valued corner in the top-left position, and the lowest adjacent side value in the top-center position.)

There are 4 possible solutions for the minimum possible common sum : ( Read as :a1, a2, a3, a4, a5, a6, a7 , a8 , a9 , a10 , a11 , a12 , a13 , a14 , a15 , a16 , a17 , a18 , a19

)

The lowest possible common sum = 95

7,29,59,13,23,61,11,47,37,41,17,71,83,31,5,73,67,53,79

7,29,59,17,19,53,23,61,11,37,47,41,83,31,5,43,71,79,67

7,41,47,19,29,13,53,31,11,67,17,71,83,43,5,73,61,79,37 ( This one is Jean Charles Meyrignac's solution )

7,41,47,37,11,61,23,13,59,19,17,71,83,43,5,73,79,31,67 ( This one is Pierre Audi's solution )

The lowest possible prime common sum = 101 The only solution is :

5,79,17,43,41,29,31,59,11,83,7,89,73,61,23,71,37,67,47

If all primes are below 100 , the highest possible common sum = 187 The only solution is :

37,71,79,41,67,47,73,31,83,43,61,89,53,11,97,29,23,7,17

Jean Charles Meyrignac's solution of the hexagon with the smallest consecutive primes ( Sum = 223 ) does not have 19 consecutive primes . ( His solution has 113 but not 109 , which I believe is a prime ).

So the following question is still open:

What is the lowest possible common sum if all the 19 primes are consecutive primes ? ( Can it be done ? )

I have checked out all sets of consecutive 19 primes starting from ( 3 , ... , 71 ) to ( 4597, ... , 4751 ) , and have not yet found a solution . So , my guess is that it can't be done . But , who knows ?? So, I'm still running my program to check up primes below 10000 .

Finally , regarding the non-prime 1-19 hexagon , where C = a15 = the central integer , and S = the common sum , here are my results :

C = 2

S = 22

1,9,12,6,4,11,7,10,5,14,3,18,19,8,2,17,16,15,13

1,13,8,10,4,7,11,6,5,14,3,18,19,12,2,17,16,15,9 ( Also found by Jean Brette )

1,14,7,6,9,5,8,10,4,15,3,18,19,13,2,17,11,16,12

S = 24

3,6,15,1,8,12,4,11,9,10,5,16,19,7,2,17,14,13,18

S = 25

4,10,11,9,5,14,6,3,16,1,8,13,19,12,2,15,18,7,17 ( Also found by Jean Brette )

4,12,9,11,5,3,17,1,7,10,8,13,19,14,2,15,18,16,6

S = 27

6,11,10,4,13,5,9,1,17,3,7,14,19,15,2,18,12,8,16

S = 29

8,4,17,1,11,3,15,5,9,6,14,7,19,10,2,13,16,18,12

8,6,15,5,9,3,17,1,11,4,14,7,19,12,2,13,18,16,10 ( Also found by Jean Brette )

C = 4

S = 22

1,18,3,9,10,7,5,11,6,14,2,19,17,15,4,16,8,12,13

S = 24

2,10,12,1,11,7,6,13,5,16,3,19,18,8,4,17,9,15,14

2,16,6,13,5,7,12,1,11,10,3,19,18,14,4,17,15,9,8

S = 25

2,7,16,1,8,14,3,12,10,9,6,17,19,5,4,15,13,11,18 ( Also found by Jean Brette )

2,17,6,9,10,1,14,3,8,12,5,18,19,15,4,16,11,13,7 ( Also found by Jean Brette )

S = 26

3,15,8,2,16,1,9,7,10,11,5,18,19,14,4,17,6,12,13 ( Also found by Jean Brette + There is a typo ; you missed out 8 )

S = 28

5,7,16,1,11,2,15,3,10,12,6,17,19,8,4,18,13,14,9

5,12,11,1,16,2,10,3,15,7,6,17,19,13,4,18,8,9,14 ( Also found by Jean Brette )

S = 29

6,12,11,2,16,5,8,3,18,1,10,13,19,14,4,15,9,7,17 ( Also found by Jean Brette )

S = 31

9,3,19,1,11,7,13,6,12,2,17,5,18,8,4,10,16,15,14 ( Also found by Jean Brette )

8,5,18,3,10,6,15,2,14,1,16,7,19,9,4,11,17,13,12

C = 6

S = 23

1,12,10,5,8,11,4,17,2,18,3,19,16,7,6,14,9,15,13

1,18,4,17,2,11,10,5,8,12,3,19,16,13,6,14,15,9,7 ( Also found by Jean Brette )

S = 24

1,10,13,8,3,19,2,18,4,9,11,12,17,5,6,7,15,14,16

1,12,11,10,3,8,13,9,2,18,4,19,17,7,6,14,15,16,5 ( Also found by Jean Brette )

S = 27

4,15,8,10,9,11,7,1,19,3,5,18,17,13,6,16,12,2,14

3,5,19,1,7,9,11,12,4,15,8,16,18,2,6,13,14,17,10 ( Also found by Jean Brette )

3,5,19,1,7,16,4,13,10,8,9,15,18,2,6,12,14,11,17 ( Also found by Jean Brette )

S = 33

10,4,19,1,13,2,18,3,12,5,16,7,17,8,6,11,14,15,9 ( Also found by Jean Brette )

10,5,18,3,12,2,19,1,13,4,16,7,17,9,6,11,15,14,8

C = 8

S = 24

1,17,6,16,2,19,3,9,12,7,5,18,15,10,8,11,14,4,13 ( Also found by Jean Brette )

S = 25

1,14,10,3,12,9,4,19,2,17,6,18,16,7,8,11,5,15,13

1,18,6,17,2,9,14,4,7,13,5,19,16,11,8,12,15,10,3 ( Also found by Jean Brette )

S = 26

2,10,14,1,11,9,6,17,3,18,5,19,16,4,8,13,7,15,12 ( Also found by Jean Brette )

2,18,6,17,3,9,14,1,11,10,5,19,16,12,8,13,15,7,4 ( Also found by Jean Brette )

1,10,15,7,4,9,13,11,2,18,6,19,17,3,8,12,14,16,5

S = 27

3,18,6,4,17,1,9,11,7,15,5,19,16,13,8,14,2,12,10 ( Also found by Jean Brette )

2,16,9,11,7,15,5,4,18,3,6,19,17,10,8,13,12,1,14 ( Also found by Jean Brette )

1,10,16,9,2,19,6,14,7,5,15,11,18,3,8,4,17,12,13

1,11,15,3,9,16,2,19,6,7,14,12,18,4,8,5,10,13,17 ( Also found by Jean Brette )

1,11,15,10,2,9,16,5,6,14,7,19,18,4,8,12,17,13,3 ( Also found by Jean Brette )

S = 28

1,10,17,7,4,9,15,11,2,12,14,13,19,3,8,6,16,18,5

1,12,15,11,2,9,17,7,4,10,14,13,19,5,8,6,18,16,3 ( Also found by Jean Brette )

S = 31

4,15,12,3,16,2,13,1,17,5,9,18,19,11,8,14,7,6,10 ( Also found by Jean Brette )

C = 10

S = 28

2,12,14,9,5,6,17,8,3,18,7,19,16,4,10,11,13,15,1 ( Also found by Jean Brette )

S = 29

2,9,18,4,7,19,3,15,11,5,13,14,17,1,10,6,12,8,16 ( Also found by Jean Brette )

1,11,17,8,4,19,6,9,14,3,12,16,18,2,10,7,15,5,13 ( Also found by Jean Brette )

S = 31

3,9,19,4,8,17,6,11,14,1,16,12,18,2,10,5,13,7,15

2,11,18,6,7,15,9,5,17,1,13,16,19,3,10,8,14,4,12

S = 32

3,12,17,2,13,1,18,8,6,11,15,14,19,5,10,7,9,16,4

C = 12

S = 29

3,15,11,2,16,5,8,17,4,18,7,19,14,6,12,10,1,13,9

S = 32

3,10,19,7,6,8,18,9,5,11,16,13,17,1,12,4,14,15,2

3,11,18,9,5,8,19,7,6,10,16,13,17,2,12,4,15,14,1

S = 33

5,9,19,8,6,13,14,1,18,4,11,17,16,2,12,10,15,3,7

4,10,19,9,5,15,13,6,14,1,18,11,17,2,12,3,16,7,8

4,11,18,10,5,9,19,1,13,6,14,15,17,3,12,7,16,8,2

3,16,14,2,17,1,15,5,13,9,11,19,18,7,12,10,4,8,6

2,16,15,5,13,9,11,4,18,1,14,17,19,6,12,7,8,3,10

S = 34

6,10,18,1,15,2,17,3,14,11,9,19,16,4,12,13,7,8,5

6,11,17,3,14,2,18,1,15,10,9,19,16,5,12,13,8,7,4

5,10,19,1,14,2,18,9,7,11,16,13,17,3,12,6,8,15,4

S = 35

8,11,16,1,18,3,14,2,19,6,10,17,15,7,12,13,5,4,9

6,11,18,3,14,2,19,1,15,7,13,16,17,5,12,10,9,8,4

S = 36

8,11,17,1,18,4,14,3,19,2,15,13,16,7,12,9,6,5,10

C = 14

S = 27

1,16,10,13,4,15,8,17,2,18,7,19,12,3,14,6,9,11,5

1,18,8,17,2,15,10,13,4,16,7,19,12,5,14,6,11,9,3

S = 33

1,15,17,4,12,5,16,8,9,11,13,19,18,2,14,6,7,10,3

1,15,17,5,11,12,10,7,16,4,13,19,18,2,14,6,8,3,9

1,17,15,2,16,5,12,10,11,9,13,19,18,4,14,6,3,8,7

S = 36

7,10,19,8,9,11,16,2,18,1,17,12,15,3,14,5,13,4,6

7,11,18,2,16,1,19,8,9,10,17,12,15,4,14,5,6,13,3

S = 37

10,8,19,1,17,2,18,3,16,9,12,15,13,4,14,11,6,7,5

10,9,18,3,16,2,19,1,17,8,12,15,13,5,14,11,7,6,4

C = 16

S = 29

2,15,12,13,4,19,6,18,5,14,10,17,11,1,16,3,9,8,7

1,17,11,15,3,18,8,14,7,13,9,19,12,2,16,4,10,6,5

S = 31

2,17,12,15,4,18,9,8,14,7,10,19,13,3,16,5,11,1,6

S = 32

4,13,15,3,14,8,10,17,5,18,9,19,12,1,16,7,2,11,6

4,18,10,17,5,13,14,3,15,8,9,19,12,6,16,7,11,1,2

S = 34

4,18,12,5,17,2,15,9,10,13,11,19,14,6,16,7,1,8,3

S = 35

6,17,12,8,15,2,18,3,14,11,10,19,13,7,16,9,4,5,1

4,13,18,3,14,11,10,8,17,6,12,19,15,1,16,7,5,2,9

S = 36

8,10,18,1,17,4,15,7,14,13,9,19,12,2,16,11,3,6,5

8,13,15,7,14,4,18,1,17,10,9,19,12,5,16,11,6,3,2

S = 38

10,11,17,2,19,1,18,6,14,9,15,13,12,5,16,7,3,8,4

C = 18

S = 31

3,16,12,13,6,14,11,15,5,17,9,19,10,1,18,4,7,8,2

3,17,11,15,5,14,12,13,6,16,9,19,10,2,18,4,8,7,1

S = 33

3,17,13,6,14,9,10,16,7,15,11,19,12,2,18,4,1,8,5

S = 35

4,17,14,6,15,11,9,10,16,7,12,19,13,3,18,5,2,1,8

3,17,15,9,11,8,16,7,12,10,13,19,14,2,18,4,6,5,1

S = 36

5,14,17,4,15,10,11,9,16,8,12,19,13,1,18,6,3,2,7

S = 38

11,14,13,6,19,2,17,5,16,10,12,15,9,7,18,8,1,4,3

9,13,16,10,12,7,19,2,17,6,15,14,11,4,18,5,8,3,1

8,11,19,2,17,6,15,10,13,9,16,14,12,1,18,4,3,7,5

Does Sudipta is missing something?

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