# How many decimations equals a “decimation”?

You and your fellow legionaries have just conquered a neighboring village, and the Prelate has ordered per tradition that the survivors be decimated and the victims' heads delivered to him in a sack as proof of the deed.

Sadly, he's a little new on the job (and more than a little bloodthirsty), and incorrectly believes that "decimate" means to reduce the population to 1/10th its original size. Your Captain, on the other hand, is a seasoned veteran and he knows that it really only means to reduce the population by 1/10th.

Unfortunately, these two officers have very limited means of communication. The Prelate can only communicate with the Captain by sending an order to decimate the survivors, and the Captain can only communicate with the Prelate by sending him a sack full of heads.

Since the Prelate knows exactly how many villagers there were to start with, and will not be satisfied until his definition of "decimation" has occurred, how many times will he have to give the order before his expectations are met?

(Note: I'm looking for a mathematical answer. You could just do this on a calculator but that's not how this is intended to be solved. You may assume all rounding is to the nearest integer, though it shouldn't make much of a difference.)

• Can I ask for an explanation of the close votes? This seems to be in a similar vein to other questions I've seen here. I suppose it could have worked in Mathematics.SE, but there's plenty of other mathematics questions here as well. – Darrel Hoffman Feb 7 '16 at 19:39
• I did not vote to close, but am curious about your view that this IS a puzzle. The answer you accepted is a fairly straightforward application of mathematics, so I see it more as a very flavorful word problem (and I did enjoy the story, by the way!) I think a math problem is more of a puzzle when it requires the reader to notice a "trick" to set up the problem to be a manageable task. I am curious what solution you had in mind. Could you perhaps post it as an answer? – Regan Feb 8 '16 at 18:24
• Math puzzles should require some kind of clever insight to solve. Yours does not require cleverness, as shown by the accepted answer (I believe you have a clever, non-mathy solution, but if an easy solution exists, no one will be motivated to look for the clever one). Also, people want the premise or result to be counter-intuitive or surprising. It's not too surprising that its possible to reduce a number by 10% repeatedly until only 10% remains, and not surprising it takes about 20 reductions. See this meta post. (I didn't close vote). – Mike Earnest Feb 8 '16 at 19:10

Aprox 22.

Each time a decimation (regular) occurs, you reduce the size of the population to 9/10 of the original. So, our formula is $(\frac{9}{10})^x = \frac{1}{10}$

Taking the log of each side, bringing down the X and dividing gives $x = 21.85$

• Simpler than my solution, but effective. – Darrel Hoffman Feb 7 '16 at 18:57