### Maths

As a youngster, a friend of mine owned a Datsun 240K automatic. Some of his friends had V8s. He discovered however that he could do some impressive burnouts by selecting reverse at highway speed & flooring it!

The argument is over the relative speed of one back tire spinning, compared to the road below.

The following are given:
1. Vehicle speed 120 km/h.
2. Engine at redline, consulting gearbox ratios extrapolates this to 100 km/h in reverse.
3. Open diff, so one tire does not slip at all, but remains in contact with the road while the other spins backwards.

So, does anyone have an idea what the relative speed difference was under those conditions?

His view, by assuming the same ratios as when driving at slow speed in reverse, results in a rear wheel speed equivalent to something over 400 km/h.

My view goes something like this. First, it seems inescapable that if the gearbox engages immediately and if there is no slippage in the torque converter, then the instantaneous situation would be that the engine must be rotating backwards. That simply isn’t possible, so we have to assume major slippage in the transmission.

Assuming there is slippage even for a brief period, then where is it? The torque converter is by far the best candidate. We know that a heavy load can switch a torque converter out of coupling mode (that happens if you tow something too heavy in too high a gear). So we get the following sequence of events.

1. Reverse gear engaged in gearbox.
2. Front shaft of gearbox reverses, breaks torque converter out of coupling mode (stator locks).
3. Engine goes to full power, peak revs.
4. Torque converter passes magnified engine torque back to gearbox.
5. Gearbox passes torque (reverse direction) back to diff and to wheels.
6. Torque exceeds limiting friction of one tyre, wheel slows down and may reverse.
7. Steady state is reached when torque transmitted by the engine is equal to torque transmitted from two wheels, one slipping.

If this is the situation, then it’s not possible to predict exactly what the slipping wheel is doing, but I would guess rotating slowly forwards (same direction as road).

Here are two puzzles, superficially similar, but different answers.

A. Alf meets Bert, and asks him how many children he has, and of what sex.
The puzzle: if Bert has two children and at least one of them is a boy, what are the odds that he actually has two boys?

B. Alf now asks the same questions of several more people, until he gets to Charlie.
The puzzle: if Charlie has two children and at least one of them is a boy, what are the odds that he actually has two boys?

C. Alf goes back to Bert, and asks him what days any boys were born on.
The puzzle: if Bert has two children and at least one of them is a boy born on a Tuesday, what are the odds that he actually has two boys?

D. Alf now asks the same questions of several more people, until he gets to Dave.
The puzzle: if Dave has two children and at least one of them is a boy born on a Tuesday, what are the odds that he actually has two boys?

This strange puzzle and/or variants of it was originally set by Martin Gardner. The surprising thing is that the exact same question can have two different answers depending on the assumed context from which the odds should be calculated. There are enough clues here to make the answers fairly obvious.