Puzzling Visual Maths For The Curious Minded #5  
     
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Puzzling Visual Maths

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Resistors That Self-Replicate Themselves Button Forward
 

Self-replicating resistorsHere 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.

Thus, 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.

 

self-replication resistors 2

 

This 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?

Self-replicating Resistors 3To 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).
Remarkable cases:
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…

*Source: Problem of the Week 1165 by Stan Wagon, Macalester College. Read more: http://www.futilitycloset.com/?s=self-replicating+resistors

 

 

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