Every family has its own traditions. Mine has some that are admittedly… nerdy. But we love them just the same. One of them involves birthdays, and since today is my birthday (I’m 31… the “prime” of my life!) I thought I’d share it with you!
When you’re young, it’s simple: you get one candle for every year of the birthday you’re celebrating and all the candles tend to match. As you get older, the practicality of putting one candle per year decreases, and the candles that do make it to the cake are typically what’s left over in the drawer. At some point my family took the hodgepodge of candles as a mathematical challenge: what value could each color of candle take so that the sum equal’s the birthday person’s age?
As we got older that became too simple, so we added a twist that I have lately come to realize is exactly the kind of challenge that the GMAT test makers love to give: let each color of candle represent a unique prime number such that the sum equals the number of years being celebrated.
Here’s an example to get you started for my 31st birthday:
One red, two green, three purple = 31
1. Remember that 2 is the only even prime number. Whether you need to make an even or odd total will tell you how/if you can use 2.
2. Start with the small primes. A few small primes and one big one is usually the best approach.
Red = 19
Green = 3
Purple = 2
Here’s what makes this such a great GMAT problem. It touches heavily on primes and number properties: two topics that the GMAT loves because they’re more focused on your pattern recognition and problem solving than your math skills. Let’s look at how logically applying those skills gets you to this solution.
First of all, we need Total Red + Total Green + Total Purple = 31. Let’s look at that in terms of evens and odds. Total Red + Even + Total Purple = Odd. Since there are two green, no matter what value we put in that spot, the total will be even. So, moving some things around we get the following:
Total Red + Total Purple = Odd – Even
And, since Odd – Even is always Odd, we get Total Red + Total Purple = Odd.
If the sum of two integers is odd, that always means that one of them is even and the other is odd. We have an odd number of red and purple candles, and the only way to get an even product from Odd* Integer is for that integer to be even. That’s where our use of 2 comes in. Since there is only one even prime number, two must be the value for either red or purple. Once we plug that in we only need to test a few values until we find a combination that works! (There may be others… list them in the comments if you find a good one)
Here’s a few celebrity birthdays for March 24th and their candles. See if you can figure these out! I’ll post suggested solutions at the bottom, but feel free to find your own!
Jim Parsons 41- One yellow, three red, two blue, two green
Peyton Manning 38- Four red, two green
Chris Bosh 30- One yellow , one blue, one green
Alyson Hannigan 40- Two pink, two orange
And the challenge question… today Harry Houdini would have turned 140- One blue, one red, two green
Jim Parsons 41- One yellow (19), three red (2), two blue (5), two green (3)
Peyton Manning 38- Four red (7), two green (5)
Chris Bosh 30- One yellow (2), one blue (5), one green (23)
Alyson Hannigan 40- Two pink (7), two orange (13)
Harry Houdini140- One blue (5), one red (131), two green (2)