Space is hard on equipment. Very hard. Take temperatures: On the surface of Mars, they can range from a summer's balmy 81 degrees Fahrenheit (27 degrees Celsius) to –225 degrees F (–143 degrees C). For a point of comparison, the coldest naturally occurring  temperature ever recorded on Earth was –128 degrees F (–89 degrees C). Add to this the harmful radiation from space (which the Martian atmosphere does not screen out) and you will understand why the survival of the two Mars rovers for five years is so remarkable.

In fact, the way most spacecraft achieve survivability is through triply redundant equipment. If something breaks, there are two spares to take its place.

The trouble is that spares mean more weight, more expense and less science. What if the redundancy worked differently? What if a backup could do several functions instead of just one? Each function requires full-time work, so once a backup is engaged, it does only one thing.

Warm-Up Puzzle:

Suppose you have six primary devices, each serving a different function. You also have three backups:
Backup 1 can do functions A, B, C, D.
Backup 2 can do C, D, E, F.
Backup 3 can do A, B, E, F.

How many failures could you tolerate among the primary devices before you could no longer do a function?

Solution to Warm-Up Puzzle

Notice that if the backups could handle only one function each, then six would be needed—that is still assuming that only the primaries fail. But what if backups could fail, too?

Now here are problems for you:

1. If each backup could do only one function, then how many backups would be necessary to tolerate two failures of primaries or backups?

Solution to Problem Number 1

2. In the more versatile setting of the warm-up, assume that both backups and primaries could fail. How many failures could be tolerated with the system still performing all functions?

Solution to Problem Number 2

So, our system can do as well with three versatile backups as with 12 nonversatile ones. Not bad.

3. Suppose the backups remain as in the warm-up, but the primaries are also flexible. That is, the primaries that are currently doing A and B are as flexible as backup 1. The primaries that are currently doing C and D are as flexible as backup 2. The primaries that are currently doing E and F are as flexible as backup 3. It is conceivable, then, that a primary (P1) fails and that a backup B takes over another primary's (P2's) function while P2 does P1's function. In that case, how many failures can be tolerated?

Solution to Problem Number 3

4. How many backups would be needed if each primary and backup could do only one function to tolerate the same number of failures?

Solution to Problem Number 4