You have a lot of identical books, stacked on the edge of a table. How far from the table-edge can the pile reach without falling over? This is an important issue in construction, for example in building arches above entrances.
No-one submitted a prize-winning entry. Alex On and Khaleah Edwards deserve honourable mention for getting most of the way there, but neither wrote it up.
Break the problem down into two smaller problems by investigating how much further (if further at all) the pile of books can project if it’s n+1 books rather than n books.
Suppose for simplicity each book is of length 2.
Suppose you have n books stacked to reach out as far as it is possible with n books.
Consider those n books as a single block, and imagine the (n+1)th book being inserted below that block.
The pile of (n+1) books will still balance if: the block of the top n books is moved a distance 1/ (n+1) out from the edge of the table, and the bottom [(n+1)th] block is positioned so that its centre is n/(n+1) in from the edge of the table.
Why? Because moving the block of the top n books out 1/(n+1) unit moves the centre of mass out n/(n+1) units. Having the bottom block so that its centre is n/(n+1) units in from the edge of the table moves the centre of mass of the whole pile in n/(n+1). The two things balance each other out, so if the centre of mass was previously just a tiny bit on the right side of the edge of the table, then it still is.
So the pile can reach 1/(n+1) units further out if it’s (n+1) books rather than n books.
You’ve proved that if you go from n books to n+1 books, the pile of books can project a bit further out.
However, it could still be the case that the distance the pile projects out is limited.
If you are hungry and eat one whole pizza; then you’re still hungry and eat a 1/2 pizza; then still a bit hungry and eat a further 1/4 pizza; then still a bit hungry again and eat a further 1/8 pizza; and so on for ever – then at each stage the amount you eat increases, but the total amount is always less than two whole pizzas. An infinite sum can have a finite total.
Is this the same?
In fact, it’s not. As n increases, the total reach of the pile of books not only increases, but increases without limit.
Why? From Step 2, the formula for the total reach is
1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + …..
Divide that addition up into successive chunks: (1/3+1/4), (1/5+1/6+1/7+1/8), (1/9+1/10+…+1/16), (1/17+1/18+… +1/32). etc. Each chunk is longer than the previous one (includes more terms), but each chunk adds up to bigger than 1/2.
So the total reach is bigger than
1 + 1/2 + 1/2 + 1/2 + 1/2 + ….
which means that in theory it is bigger than any number you can imagine. It’s infinite.
In practice the limit is smaller. As the pile gets higher, the centreof gravity also gets higher, so an even tinier wobble or vibrationwill make the pile fall over. But you can get a surprisingly big reach even in practice: