I love this! Wish I could be in NY for the main event this summer. I voted for Dad, but only 'cause I am closer to his age than teen ti's
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Assuming one drives ~1500 miles per month and gets 25mpg, your wager is 60 gallons of dinosaur juice. At ~$4/gal, the wager is ~$240. Let's call it $250 to account for the estimation factor.
More so than wheel size, my experience is that the right tires on a lighter car make the biggest difference for an autoX. So, Joe... I think you need to slap $600 worth of 225/50-16 competition rubber on the stock 16-inch wheels that came with your son's ti. Inflate them to ~45lbs at each corner to increase the stickiness, dial out any understeer and keep the bottom of the tire in contact with the pavement. You will drive circles around him!
teen ti will then subsidize your outlay by buying one month's worth of gas for you. You will be out $350 for the win, but you will corral eternal bragging rights to being the better driver. Isn't that worth $350?
When it is all over, you might even be able to sell him the new sticky autoX wheel/tire setup for $350! $350 for a set that has only been used for one autoX is a deal! I'm sure there are many autoX'ers here that will buy them from you if he doesn't. Bwahahahahah! Age and experience, indeed! [Joe--Hope I did not just give away your strategy!]