Before I post this question, let me start by saying that I **love** a good bet. My husband and I bet on everything — that is the only way chores get done in the house. And, FYI: he is currently on 2 weeks of “diaper duty” for our 1-year-old since he lost 2 bets! Don’t mean to rub it in or anything 😉 We are poker players… So here it comes:

**Q: **

**I have a math question that I hope can help solve a $5 bet I have with my friend Steve. It’s not advanced math by any means but Steve and I can’t seem to agree on who is right. First, a little background:**

**Our local driving range sells a medium bucket of balls for $7 and $9 for a large bucket. They offer a nice discount if a person buys a “punch card” that is good for 10 buckets of balls. Steve and I each pay $70 for a card which is good for 10 large buckets of balls. It’s a $20 savings for each of us! Another perk is if someone saves their cards and turns in 10 punched out cards (having spent $700) they will receive a free card worth, in our case, 10 more large buckets worth $70. Pretty cool!**

**Here’s what happened: Steve had saved 7 punched out cards and went to buy another. After he paid $70 for his 8th card and mentioned that he only needed to buy 2 more cards to be eligible for his free card some guy who was near us and whom we didn’t know said “Hey, I’m moving out of town so I won’t need this punched out card that I’ve been saving” and he gave it to Steve.**

**As we were walking out to hit some balls I said to Steve, “Wow, that guy saved you $70 towards your free card”. Steve disagreed. His reasoning was “what if I found 10 punched out cards that someone had lost and I turned them in to get my free card”? He said the free card is worth $70 so by dividing the number of found punched out cards (10 cards) into the value of the free card ($70) then each punched out card was only worth $7. To Steve the card given by that unknown guy saved Steve only $7. I told Steve that because of that gift card given to him he would end up paying only $630 instead of $700 to get his free card, therefore that guy saved him $70.**

**So, can you explain who wins the $5?**

A: This is a more difficult question than meets the eye!

OK. So Steve is not getting a punch card that is worth $70 (because it is all used up)… So let’s analyze this and let me create a scenario:

REGULAR SCENARIO

For $700 you essentially get 11 punch cards (the ten paid for + 1 free)

Price per punch card = $700/11 = $63.64

SCENARIO A

Steve has 9 punch cards = $630 spent.

A guy gives him a free punch card (already used)….

Steve can now get an unused free punch card.

Now Steve has had use of 10 punch cards for $630.

Price per punch card to Steve: $630/10 = $63.

SCENARIO B

Steve has 8 punch cards = $560 spent.

A guy gives him 2 free punch cards (already used)….

Steve can now get an unused free punch card.

Now Steve has had use of 9 punch cards for $560.

Price per punch card to Steve: $560/9 = $62.22.

SCENARIO C

Steve has 7 punch cards = $490 spent.

A guy gives him 3 free punch cards (already used)….

Steve can now get an unused free punch card.

Now Steve has had use of 8 punch cards for $490.

Price per punch card to Steve: $490/8 = $61.25.

So, notice that the used punch-card reduces your price per card… but not by much. The only “fair” comparison in my mind would be to compare what the cost was to play the same number of buckets — but this is a very hard question because:

- 11 buckets in the first scenario cost $700 (but you have no progress on your next free card)

- 11 buckets in the scenario A cost $630 for 10 + $70 for 1 = $700 (but you have a 1-card progress toward the free card)

- 11 buckets in the scenario B cost $560 for 9 + $140 for 2 = $700 (but you have a 2-card progress toward the free card)

……….

See the difficulty with this? The exact “savings” is a little fuzzy.

My dad brought up an interesting question to help us realize how difficult of a “bet” you made :)…. What are Steve’s intentions?

Let’s say he decided that morning that he had 9 punch cards in his pocket and he was going to buy 2 more punch-cards of golf and then retire from golf for life…. To buy his 2 punch-cards would cost him $70 ($70 for the 10th card + 1 free). Now, let’s say some guy handed him a used punch-card (as happened)… How much would 2 punch-cards cost him? Well, he would get the free one automatically, but then he would have to pay $70 for the secord card. In both scenarios, he is paying $70 for an additional 2 cards. No savings!!!

OR, let’s say he had decided that morning that he was going to buy 1 more punch card and then retire for life (and not even play the free one he would have received). The used punch card gives him his 1 last card for free, saving him the $70 he was planning on spending. A $70 savings!

See how the intentions matter?

The benefit of receiving the used punch card is really that it helps you toward progress on your next card…. it’s long term savings is small…. In the long run, if you play golf multiple times a week for the rest of your life, the used-punch card saves you pennies, if that…

I am going to give this problem more thought and get back to you if I get any new insight… but for now:

Who wins? The only fair answer would be no one.

So, when no one wins a bet, it is time to “let-it-ride”. Put that $5 on one drive: longest drive takes all.

Hi Stacey,

Wow, you really spent some time on this!! My basic thought was that the driving range guys needed some kind of proof that SOMEONE had spent $700 and proved it by handing in 10 punched out cards. It didn’t matter to them if those cards were all bought by the same person or if some of them were found in a trash can or some were simply gifts. So, for every 10 punched out cards turned in it proved that $700 had gone into their cash register and they were willing to hand a free $70 card to whoever turned in those 10 used cards.

Because of the free “gift card” given by that stranger my friend Steve was given that free 11th card ($70 value) after spending only $630 and not $700. That’s how I came up with the $70 savings.

Stacey, I can’t thank you enough for spending so much time on this silly bet. If you and your husband ever travel through San Clemente or if my wife and are ever in your area it will be a free dinner on us. Thank you!!!

Mike,

I was quick to say that you were right (when I first thought about it), but then realized.. Normally your friend would get 11 cards for $700. In this case, he got 10 cards for $630 (the 11th one was paid for AND used by someone else)… That is why things get so tricky!

Of course you are right in the eyes of the golf-shop. Someone paid the full $700, who cares if it was broken up.

No bet is a silly bet 🙂 Not in my family. We bet on the most ridiculous things and someone ends up paying for it (dishes, laundry, cat litter duty).

I had to look up where San Clemente is… I knew it was in California, just didn’t know which part. My husband is currently at a conference in San Francisco (and is originally from Fort Bragg, CA)…. Not that all of this is relevant, just thought I’d share 🙂

If you are buying dinner, this bet may end up costing way more than $5 😉