Professor Stewart's Hoard of Mathematical Treasures (19 page)

If the boxes are 1 unit long, the top one pokes out 11/12 of a unit.

‘How did you figure that out?’ Luigi asked.

‘Well, I put the top one on the second one, so that its centre was poised exactly on the edge. So it poked out
a unit. Then it was obvious that the centre of mass of the top two boxes was in the middle, so I placed them with the centre of mass exactly over the edge of the third box. If you do the sums, that makes it poke out another
of a unit. Then I placed the three of them so that their combined centre of mass was right on the edge of the table, and that turned out to add a further
of a unit to the overhang.’

‘And
,’ Luigi said. ‘You’re right, it does poke out almost 1 unit.’

Alert readers will observe that Angelina and Luigi are assuming the boxes are identical and they are uniform, that is,
the mass is evenly distributed. Real pizza boxes, full or empty, are not like that, but for this puzzle you should pretend that they are.

‘What happens if you add more boxes?’ Luigi asked.

‘I think the pattern continues. I could replace the table by a fourth box, and then slide the pile out until it is just about to topple, adding a further
to the overhang. Then the top box does poke out over the edge of the table: the overhang is
. And with more boxes still, I could do the same again, adding
, and so on.’

‘So you’re saying,’ said Luigi, ‘that with n boxes you can get an overhang of
units. Which I instantly recognise as
H
_n
, where
H
_n
is the
n
th harmonic number:

Isn’t that right?’

Angelina agreed that it was. As you do.

This is a time-honoured puzzle, and the biggest overhang you can get with n boxes using this method is indeed
H
_n
, so Angelina and Luigi are right. You can find the details nicely worked out in many sources, and I would have included them, but for one thing: this traditional answer is valid only with the extra assumption that exactly one box occurs at each level. And that raises a very interesting question: what happens without that assumption?

In 1955, R. Sutton noticed that, even with just three boxes, you can do better than Angelina: an overhang of 1 instead of
. With four boxes, the biggest possible overhang is