That Self-Replicate Themselves
is an interesting life application of self-reference.
The positive consecutive integers 1, 2, 3 and 4 are
the values of four resistors that form a self-replicating
set. Opposite you can see four electrical networks, each
of which consists of the four resistors 1, 2, 3, 4 used
once and only once, their resistance values being 1, 2, 3, 4.
any one of the numbered resistors in these networks can
be replaced by one of the networks themselves, as shown
below. This replacement operation can be repeated infinitely
creating a kind of fractal or Droste effect. This set of
integer arrangements having such properties is known to
be unique up to some obvious symmetries (such as scaling).
It is also known that there is no integer solution for
networks with only 2 or 3 resistors.
leads to an interesting problem. Suppose two distinct sets
of positive integers a, b, c, d and w, x, y, z (with a ≠ b ≠ c ≠ d ≠ w ≠ x ≠ y ≠ z)
are assigned once to each electrical network in its respective
set and that the two distinct sets of networks are co-replicating
in the following sense: the four electrical networks of
one set (on the left) can be used to replace the four resistors
in any of the networks of the other set (on the right),
and vice versa. Will you then be able to find the resistance
value of each resistor in each network?
solve the puzzle you will need to know that the total resistance
of m and n in ‘series’ is
simply m + n,
and if they are placed in ‘parallel’ it is m || n or
1/(1/m + 1/n) = mn/(m + n).
m || n if n = m,
then the total resistance will be m/2;
m || n with n = m/2,
the total resistance will be m/3;
m || n with n = m/3,
the total resistance will be m/4;
m || n with n = m/4,
the total resistance will be m/5;
and so on…
Problem of the Week 1165 by Stan Wagon, Macalester College.
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