More than a century ago, during a lecture in Baltimore, Lord Kelvin asked the following rhetorical question: "Of all the two hundred billion men, women, and children that have walked across wet sand from the beginning of time down to the meeting of the British Association in Aberdeen in 1885, how many would answer anything but 'yes' to the question: 'Did the sand become compressed under your foot?' " Why Aberdeen in 1885? That was where O. Reynolds showed that the sand actually expands rather than contracts under our feet, contrary to common sense.
But let's not digress too far from our subject. It would be better to ask a similar question about the emperor's award: "Of all the millions of readers of Perelman's book and the thousands of Quantum readers, how many noticed that the behavior of both the emperor and the general described in the story was at least strange and absolutely illogical!"
What was so strange and illogical? We'll soon see.
First, let's try to estimate some of the values we'll need. It's clear from the story that a coin with a mass of 655 kg was just about at the limit of Terentius's physical resources: a little more and he would be unable even to budge it. We'll estimate this "little bit" as 45 kg — that is, assume that the biggest coin that would yield to Terentius's efforts has a mass of 700 kg (which corresponds to a denomination of 140,000 brasses). In addition, assume that Terentius's state of health will allow for daily visits to the treasury and removal of new coins for ten thousand days (about 25 years).
So, the emperor decided to lure his combative general into a trap that is often called the avalanche. (Indeed, it's hard to think of a better name: the coins grow like an avalanche, and this is what the miserly and cunning emperor counted on.) In this case, the multiplication factor is k = 2 — that is, each coin is twice as massive as the previous one.
And it is this choice of multiplication factor that leads one to suspect that the emperor was a strange person, because of all positive integers k, he had chosen the one that brought the greatest profit to Terentius!
Consider, for instance, the case in which each new coin is three (and not two) times as massive as the preceding one. How many coins would Terentius be able to lift? The value of the (n + 1 )st coin would then be 3n brasses. The general can lift a coin that is equivalent to no more than 140,000 brasses. What is the largest n such that 3n < 140,000? This n satisfies the inequalities
3n < 140,000 < 3n+1, or
Log3(140,000) - 1 < n < Log3 (140,000).
Since Log3140,000 = 10.7... , we get n = 10. So the last coin Terentius would be able to lift is the eleventh: on the first day he'd receive 1 brass, on the second day 3 brasses, on the third 3^2 = 9 brasses, and so on. The total reward would come to 1 + 3 + 32 + ... + 310 = 88,573 brasses. Remember, with k = 2 he received 262,143 brasses — almost three times as many!
A similar situation would occur for larger values of A. In general, the sum S that Terentius could receive in n days given a factor k equals
S= k2+ ... +kn,
where n = [logk140,000] (and [a] denotes the greatest integer not exceeding a). By the formula for the sum of a geometric sequence,
s = (kn+1 - 1)/ (k-1)
If n is large enough, we can assume n = [Logk140,000] = Logk 140,000; then
This means that S actually decreases as k increases. In our case, however, this is not exactly true: the logarithm isn't very large, so in fact S(k) decreases "irregularly." Here are a few values of S: S(4) = 87,381; S(5) = 97,658; S(6) = 55,987,- S(7) = 137,257; S(8) = 37,499; S(10) = 111,111; S(20) = 8,421; S(50) = 127,551; S(100) = 10,101 (in the last case, Terentius would come for his reward only three times!).
And what happens for k = 1 ? Perhaps in this case the sum S turns out to be greater than for k = 2? Alas, that's not the case. Another factor comes into play here — the somber fact of human mortality. We've already estimated the time allotted to Terentius for receiving his reward as 10,000 days. Consequently, in the case k = 1, he would get simply 10,000 brasses.
Of course, we could adopt other limitations instead of 700 kg and 10,000 days. Then our conclusions would have been somewhat different—for instance, with 1,000 kg as the greatest possible mass of the coin we'd have S(2) < S(3)—but basically they would remain the same.
Thus, the (supposedly) cunning emperor, having decided to cheat the general using the avalanche effect, chose the worst multiplication factor (or at least one of the worst). And this gives us grounds to consider him a strange person—to put it mildly.
And what about Terentius? This is a little more complicated. It may help to take a leap…