If you’re like most television owners or social event attendees this weekend, you already have plans to watch the Super Bowl. And if you take a second to think about it, it’s actually kind of a dumb thing you’re about to watch. Not the football game, which as a red-blooded American I’m honor-bound to admit is thoroughly awesome, but the name itself. Super Bowl? When was the last time you called something “super” to someone over the age of 12?
“Super” is a sticker that goes on someone’s homework or cavity-free dental chart, not the brand name of one of the global economy’s largest events. But there it is, so firmly entrenched in our culture that you’ve likely never noticed until you just took the time to think about it. “Super Bowl” Why is it called that? And how can knowing the back story make help you as you approach your own personal Super Bowl in your GMAT test day?
When the National Football League and American Football League decided back in the late 1960s to play a season-ending game between each league’s champion, the unofficial working title of “AFL-NFL Championship Game” didn’t quite have that super pop to it. So as league executives brainstormed ideas, Kansas City Chiefs owner Lamar Hunt returned home one evening to find his young son enthralled by his latest cheap toy — his “Super Ball,” a rubber, bouncy ball produced by the Whammo corporation. As the boy bounded about the house yelling “Super Ball, Super Ball!” Hunt saw some synergy between “Super Ball” and the names of other high-profile football games, the Rose Bowl, the Cotton Bowl, etc. Super Ball? Super Bowl — a slight twist on an existing idea, and a marketing bonanza was born. This multibillion dollar event is named after a five-cent rubber toy, all because an executive had the ability to see an unlikely relationship between the words “ball” and “bowl.”
Because business schools are obviously interested in producing the kinds of executives who can turn five-cent investments into billion dollar brands, the GMAT will provide plenty of opportunities for you to demonstrate that you can find those unlikely relationships. Nowhere is this more evident than in the realm of Number Properties, a favorite GMAT subject for math problems. While certain Number Properties categories should quickly become second-nature for you — even/odd and unit’s digit number properties, for example, are the types that you should train yourself to specifically seek — the most challenging number properties are those that require you to recognize patterns and relationships. For those, there is really a two-phase takeaway that we’ll explore in the problem below:
1) Many large-number problems can be solved simply by looking for and extrapolating patterns that you establish using small numbers
2) You can typically find the reason that the pattern exists if you ask yourself “why;” on test day this may not be the best use of your time unless you suspect that the pattern won’t always hold, but during practice you should regularly try to ascertain the logic behind the pattern, as doing so adds quite a bit of number-theory firepower to your arsenal.
Consider the problem:
What is the tens digit of 11^13?
Here, it should be clear that you want no part of multiplying 11 thirteen times — that number will be massive and the process will be insanely time-consuming. So what are you to do? Knowing that problems involving large numbers, and particularly those involving exponents — remember, exponents are just repetitive multiplication, so patterns should emerge — tend to lend themselves to pattern-driven number properties solutions, you should try to establish a pattern using small numbers. Here:
11^1 = 11
11^2 = 121
11^3 = 121 * 11, so you’d take 121 and add 1210 to get 1331.
11^4 = 1331 *11, so you’d take 1331 and add 13310 to get 14641
And here you can take a look back to see if you can find a pattern — as of the first four terms in this series, the tens digits are 1, 2, 3, and 4. Given that the tens digit appears to be the same each time as the units digit of the exponent, it’s quite likely that the tens digit of 11^13 will be 3. On test day, you can feel pretty confident in selecting C as your answer at this point.
But especially in practice you may want to go a step further to really understand the backstory behind the pattern. And if you look at how we multiplied 11s, you may see why. Why do we simply add 1 to the tens digit each time? Well, to multiply by 11, you’re really just multiplying by (10 + 1) as we did above. Multiplying by 1 gives us the original number, and multiplying by 10 shifts all of those digits left, so for 1331*11, you’d have:
As you’ll see, since the original units digit was a 1, when we multiply by 10 that units digit of 1 shifts to a tens digit of 1 to add to the original number, so we can guarantee that we’ll continue to add 1 to each tens digit and the pattern will continue: the tens digit will always be the same as the units digit of the exponent (for bases of 11).
To summarize, the way to handle tough large-number, exponent problems is often to try to establish a pattern using small numbers. Once you’ve found the pattern, you may often be able to determine a reason to guarantee that it will extrapolate to larger numbers, but even if not you can make a reasonable determination as to whether it will hold. To train yourself to think like Lamar Hunt and the original Super Bowl marketing crew, push yourself in practice to notice those relationships, and you’ll be that much better prepared to do battle in the biggest game of your GMAT season. As you prepare for your GMAT Super Bowl, follow the pattern of the bouncing ball… the Super Ball.