## The Number Is

In my last post, I asked you to determine the next number in the following sequence:

2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 2, 2, 2, 2, 0, 0

I warned you that this is a stupid sequence, having nothing to do with actual mathematics. And I gave two hints: The numbers are related to an event that has been followed in recent weeks around the world. And, this event has unfolded over a sequence of days; associate the numbers of the sequence to the days.

I’ll now reveal the answer. If you missed the last post and want to think about the problem, read no further.

The answer is 2. In fact, here’s how the sequence continues:

2, 2, 0, 0, 1, 1, 0, 0, 1, 1

It essentially stops there. Or, you can keep going, with lots of zeroes.

What’s going on? The event that is the basis for this puzzle is the 2010 World Cup, in which 32 national teams have been competing since June 11 to determine the soccer world champion.

Let me briefly review the setup. The 32 teams were divided into 8 groups of 4. For the first two weeks, each team in a group played the other three in a round robin. This resulted in a ranking of the 4 teams in a group. The top 2 continued on; the bottom 2 went home. The 16 surviving teams were put into a single elimination bracket, with the 8 top-ranked teams meeting the 8 second-ranked teams in the round of 16. This round ended just yesterday. The 8 winners will now meet in four pairings in the quarter-finals. The winners will play in two semi-final games. The two semi-final losers will play for third place; the two semi-final winners will play for the championship.

In group play, with each team playing 3 others, 6 games are required for each group (count them up and you’ll see). Since there are 8 groups, this means 48 group games were played. On day one, June 11, two group games were played. The next day three group games were played. The next day, another three. And so on for ten days, days 2 through 11. At this point, after day 11, just last Friday, a total of 32 games had been played. Within a given group, each team had played two others and still needed to play a third. For instance, in the group containing the US, the US had played England and Slovenia, but still needed to play Algeria, while England and Slovenia had played the other two but still needed to play each other.

In the 1982 World Cup, West Germany beat Austria 1-0 in the final game of group play, a game widely considered to be fixed. (Thanks, Joel, for the reference.) Algeria had already beaten Chile in its last game within the same group, assuring them a top-two finish and advancement unless West Germany beat Austria by exactly one or two goals, in which case both West Germany and Austria would advance. Had the two games been played simultaneously, West Germany and Austria could not have colluded. Starting in 1986, the final group games are indeed played in parallel. And that’s why the 3-a-day pace of games that occurred through the earlier stages of group play came to an end last Saturday, replaced by 4 games a day, with the teams in one group playing parallel games and then, later in the day, the teams in a second group playing parallel games. Here in Seattle, the first set of games started at 7:00 AM; the second at 11:30 AM.

If you’ve followed this, you’ll see that group play consisted of a day of 2 games, ten days of 3 games, and four days of 4 games:

2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4

Then came the round of 16, single-elimination games, from Saturday to Tuesday (yesterday), two a day:

2, 2, 2, 2

The eight surviving teams rested today and rest again tomorrow:

0, 0

But these are the numbers that start our sequence! The next number in the sequence must be the number of games to be played Friday. And that is:

2

That’s our answer. On Friday, two quarterfinal games will be played and on Saturday the other two. Then two rest days, then a semi-final game next Tuesday, the other semi-final game next Wednesday, two more rest days, the third-place game on Saturday, the championship on Sunday. Putting it all together, we get:

2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 2, 2, 2, 2, 0, 0, 2, 2, 0, 0, 1, 1, 0, 0, 1, 1

Dumb, I know. But this sequence has dominated our lives here at home lately. We always know the daily number, even if we haven’t given the sequence as a whole much attention. And today’s 0 was especially noticeable, after nineteen straight days of multiple games. What will we do when 0 becomes our daily ration?

Of course, by then we’ll have the Tour de France.

## Guess the Number

You know those math puzzles where you’re given a sequence of numbers and are asked what’s next? This is one of those. There are intelligent ones and stupid ones. Intelligent ones are those that involve actual mathematics. For example, you might be given the sequence

2, 4, 6, 8

and asked for the next number. There’s nothing mysterious going on here, but one does have to know some elementary mathematics to figure out that the next number is 10.

Or, one might be given

1, 4, 27, 256

and asked what’s next. Here, one would perhaps recognize these numbers as 1 to the first power, 2 squared, 3 cubed, and 4 to the fourth power. The next number must be 5 to the fifth power, or 3125.

Stupid number sequences are those that don’t actually involve any mathematics. Here’s an example:

3, 3, 5, 4, 4, 3, 5

What’s next?

I suppose there might be other answers, but the one I’m looking for is 5. Why? Consider this sequence of words:

one, two, three, four, five, six, seven

Count the number of letters in each word. That’s the number sequence. The next number is 5 because that’s the number of letters in ‘eight’.

Okay, so here’s another number sequence puzzle, one I thought of last night, and it comes under the category of stupid number sequence. In other words, you may be able to solve it, but not by using any serious mathematics. Rather, you’ll need to associate the numbers to some other phenomenon, and if you do, the answer will be obvious. But if you don’t, there’s no hope. Ready? Here is the sequence:

2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 2, 2, 2, 2, 0, 0

What’s next?

I’ll provide the answer in a separate post so you won’t accidentally see it below, in case you want to think a little bit about it. However, let me add a couple of hints, which you can read or ignore. They are below the fold. Read more…