

A210466


Length of the shortest path to the hardest to obtain volumes in the two jugs problem.


1



2, 4, 4, 6, 6, 8, 6, 8, 8, 10, 10, 10, 10, 12, 8, 10, 12, 14, 14, 12, 14, 16, 10, 12, 12, 14, 14, 16, 16, 18, 18, 16, 16, 20, 12, 14, 14, 16, 16, 18, 18, 20, 20, 22, 22, 16, 18, 20, 22, 24, 14, 16, 20, 20, 24, 26, 26, 18, 20, 22, 22, 24, 26, 28, 16, 18, 18, 20
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OFFSET

1,1


COMMENTS

Let's enumerate tuples of numbers (x,y), with x > y, y > 1, gcd(x,y) = 1 and starting with (3,2): (3,2), (4,3), (5,2), (5,3), (5,4), (6,5), (7,2), (7,3), (7,4), (7,5), (7,6), (8,3), (8,5), (8,7), (9,2), ... .
A tuple gives the volume of two jugs. (5,3) means: a 5 liters jug and a 3 liters jug. The jugs do not have graduations. Now try to obtain 4 liters (4 is the goal) by filling (you can use an infinite supply of water) or emptying jugs or pouring a jug into the other. The shortest solution is (5,0) (2,3) (2,0) (0,2) (5,2) (4,3). Its length is 6.
All other volumes from 1 to 5 (all other goals) can be obtained in no more than 6 steps. So, 6 is the length of shortest path to the hardest goal for tuple (5,3). As (5,3) is the tuple number 4 in the list (I tried to use a "natural" order of (x,y) tuples), a(4)=6.
We only considered tuples (x,y) with gcd(x,y)=1 as they are tuples for which all goals from 1 to x can be obtained. In fact, a goal G can be obtained if G is a multiple of gcd(x,y) and G<=x.
Another example: a(14)=12. It means that, for the tuple number 14 (tuple (8,7): an 8 liters jug and a 7 liters one), the length of the solution to the hardest goal is 12. Again, 4 liters is the hardest goal, and the 12 steps to get it are: (0,7) (7,0) (7,7) (8,6) (0,6) (6,0) (6,7) (8,5) (0,5) (5,0) (5,7) (8,4).
Tuple (y+1,y) has solution length 2*(y1).  Jon Perry, Jan 30 2013


REFERENCES

Chuquet, Triparty en la science des nombres, 1484 (Jeu du tavernier).
Dudeney, Amusements in mathematics, 1917 (The Wassail Bowl).
Guyot, Récréations mathématiques, 1799.


LINKS



EXAMPLE

Table of the first terms:
x\y 2 3 4 5 6
+
3  2
4  4
5  4 6 6
6  8
7  6 8 8 10 10


CROSSREFS

Cf. A180227, A180228, A180229 (give the number of steps to obtain volume 1, 2 and 3 using any pair of jugs (x,y), x >= y >= 1 (gcd(x,y) is not necessarily equal to 1)).


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



