Change & brains

Over the last number of years, I've generally made an effort to spend my coins as I accumulate them. I don't like using a giant jar of change as a savings program, and I don't like rolling coins, and I don't like having my pockets or the top of my dresser overflowing with dirty round metal thingies.

Except for pennies. I generally don't bother carrying pennies around to try to spend them; I'll toss them in a bucket and either give them away or take them to the rip-off counting machine at the supermarket. I'll be glad when Canada finally gets rid of the penny, whenever the heck that is supposed to happen.

Over the past couple of months, I've made a change (in my behaviour), and started using cash more instead of debit for purchases, and deliberately won't spend any $1 or $2 coins. At the end of the day I throw all of my loonies and toonies in a jar. I do this because the kids are taking the bus more often these days, and being teenagers, tend not to be as organized, so now they can make change or borrow change from me when they need it. Durham Region Transit cash fare is $3, so when a fare is needed, one toonie and one loonie are removed from my jar.

I noticed at one point that my jar appeared to have rather more than toonies than loonies, and thought nothing of it other than randomness doing it's work. However, over time as the total number of coins in my jar have grown, I still saw more toonies. My mind wandered on the train ride home yesterday on this puzzle: Should I really expect my jar to have a roughly equal count of loonies and toonies?  I realized my mind had subconsciously made that assumption at some point, but was it valid? 

Well, if the things I buy with cash from day to day have a fairly random price, what would I expect? When you get change from a retailer, the value in coins you get back is between $0.00 and $4.99. Normally, you'd expect this to break downs as follows:

• For change between $0.00 and $0.99, you get 0 loonies and 0 toonies.
• For change between $1.00 and $1.99 you get 1 loonie and 0 toonies.
• For change between $2.00 and $2.99 you get 0 loonies and 1 toonie.
• For change between $3.00 and $3.99 you get 1 loonie and 1 toonie.
• For change between $4.00 and $4.99 you get 0 loonies and 2 toonies.

So, if you receive change 5 times in a day, once in each of the above scenarios, you would accumulate two loonies and four toonies. When I thought of this on the train yesterday, this blew my mind! After thinking forever that loonies and toonies were roughly equinumerous, I concluded that was way wrong. Surely we use about twice as many toonies as loonies, right? And how smart was I to figure this all out all on my own with no paper, calculator, or anything - just staring out the window on the train!

Today, a few things happened...

1. A merchant gave me $4.35 in change. She gave me 4 loonies. I may have stared at her for a moment in disbelief.
2. I googled how many loonies and toonies the Canadian Mint makes. It fluctuates from year to year, but on average the number produced for each are very similar.
3. I counted the coins in my jar. I currently have 17 loonies and 19 toonies. Eyeballing it after counting, it still looked to me that there are way more toonies than that small margin indicates.

So what have I learned?

• Barstool or train-seat logic often rests upon a zillion assumptions that are probably often wrong in a way you just didn't think of;
• Things that are bigger and shinier are probably more eye-catching to your brain;
• "In the long run" is an important concept in math & stats that needs proper respect. A few weeks and a handful of coins are not sufficiently large numbers for much of anything;
• Actually counting stuff and real data are kind of important; and most importantly:
• I am not, in fact, a genius.

On the bright side, I may think about a way to construct a future job interview question about this. I manage a small team with little turnover so haven't had a huge number of opportunities to conduct interviews, but when I have done them I've tried to throw in a few brain teasers. Not because it's that important to the particular target job that the candidate have a great deal of raw skill in math or logic puzzles, but because it's fun, and instructive to see how someone reacts when put on the spot, and see a little bit about how they think.

I like estimation questions. I've asked a couple of simple and mundane ones, but I've seen a few suggested interview questions I haven't tried yet, like "How many gas stations are there in Europe?", or "How many airplanes are in the air above Canada right now?"

In a past job competition, I asked a simple math question: "What is thirty divided by one-third?". Half the people quickly answered "Ten"; a couple took a long time and a pencil and paper to answer "Ten"; a couple actually got it right; and a couple of people completely and utterly panicked.  Fun stuff.

Years ago, when hiring for call centre positions, a colleague used to ask "Is the VCR in your living room blinking 12:00 right now?" I thought that was a perfect question for the job.