Round Numbers

Round numbers are  common, appealing, and easy to deal with. However they do not always end in a 0 or a 5. In fact, round numbers can take on all kinds of forms that most people would not think of. For example, a few round numbers in binary include 2, 4, 8, 1024, and 1,048,576. Most people would balk at these, but computer people feel quite comfortable with.

A number can be considered round in other ways as well. For example, a price of $9.99 is considered round because it has the same number across the board, and $123.45 is also round in that it is made up of a continuous sequence of consecutive numbers.

Numbers can also be round in terms of their geographic locations as opposed to their inherent magnitudes. For example, when entering a cooking time on a microwave, most people just enter an approximate time since there is usually no specific, to-the-second time prescribed. In this case, many people simply enter a round number such as 0:30 or 1:00. However 2:58 would also be round because it is a straight line down the middle of the numeric keypad, or 1:59 is a diagonal across the keypad. Even 1:06 is a round number if you enter it as 0:66. A time of 0:71 is round when entered as 1:11 and 83 seconds is round when entered as 1:23.

Most people think of round numbers as being evenly divisible by some factor (usually 5 or 10), but they can come in all manner of forms. They can come in handy too for example when negotiating a price for something. If you asked for $6,000 for something but they offer $5,000, instead of countering with $5,500, counter with $5,555.55. There’s a good chance they will accept and you will get a little more.

Perfect Game of Jeopardy!

What is the maximum amount of money that can be won on a single episode of Jeopardy!?

Show solution… ▼

No Such Thing As a Complete and Equal List

I have always had trouble with equally balanced lists; probably due to my obsessive compulsive disorder. A couple of years ago, I was auditing a University course (I believe it was a software project management course) when as usual, my mind wandered. After many years of trying to force it, it finally dawned on me that trying to make a perfectly symmetrical and complete list is literally impossible, it cannot be done, ever.

What does a complete list mean? Let’s use an example to define it. A good example is billing. When entertainers perform shows with more than one star, one of them must get “top billing”, that is, one of their names must come before the other; this is just the nature of a list. There is no way to list both names at the same time—just try to tell someone who the stars of the show are by saying both of their names simultaneously; you must say one before the other. Top billing is desirable because it implies importance. As a result, many co-stars end up arguing, even fighting over who gets top billing.

One solution is to give one top-billing, then follow it up by giving the other one top-billing. That way they each get top-billing. This does not work however because now one of them got top-billing first which is itself like saying that that person was more important, which is why they got it first.

You might propose to just balance it out by giving the second person top-billing first, in the next run, followed by the first person, and repeating, but this just ends up repeating itself. Let’s represent this with symbols:

The original list with one name listed first:

The first solution, with the second person getting top-billing, but second:

The second attempt, with the second person getting top-billing first but in the second run:

The third attempt, the second person gets top billing but again in the second half:

One more try before giving up:

As you can see, once the list has begun and one item is listed before another, there is no way that the list can be completed and symmetrical/balanced; you just get stuck in a recursive loop that grows forever. This is due to the nature of linearity. No matter what you do, A is the first item in the list and always will be! You could reverse the list so that B comes first or append the new items to the front instead of the back, but then we are in the same situation with all the symbols alternating yet never perfect:


Things get much worse with more than two symbols. The only list(s) that can be perfect are lists with only one symbol: A, AA, AAA, AAAA, and so on.

Unfortunately, there is no way to equally credit two performers, and there is no way to list two equal items. Even if you move into more dimensions there is still no way to present more than one item simultaneously.