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### Potato Puzzle

Avery tells the team about a big shipment of potato chips she got. She says “You know how they say you can’t eat just one chip? I thought I could. And I did on the first day. But each day after that I ate more; in fact, I always ate the whole number *m* more chips than the previous day. I haven’t eaten any yet today, but I realize I have eaten 1177 chips in all since I started more than a week ago.”

*How many days ago did Avery start eating the chips?*

### Cucina Combinations

At their favorite Italian restaurant, the teammates can get pizza with up to three of the available toppings (plain pizza with no toppings is allowed, but repeated toppings are not). They can also get lasagna, for which they can choose any topping (from the same list as for pizza) or no topping, for each of the two layers (bottom and top). One evening Riley realizes that there are 1,265 more varieties of pizza than lasagna that they could possibly order.

*How many different toppings does their favorite Italian restaurant offer?*

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## Solutions to week 83

**Sour Lemons**. Let *v* be the volume of the cooler in ounces. Drawing off 64 ounces and then diluting it to return the volume to *v* ounces reduces the strength of the lemonade by a factor of (*v*-64)/*v*. Coach Newton does this twice, and ends up with the lemonade at half strength, so we know that ((*v*-64)/*v*)² = 1/2. Solving, this tells us that 2(*v*-64)² = *v*² or that *v*² – 256*v* + 8192 = 0. Using the quadratic formula (and choosing just the positive root),

*v* = (256 + √(256² – 4×8192))/2 = 128 + √8192 ounces. Now 90² = 8100 and 91² = 8281, and 8192 is just about right in the middle, so it is not immediately clear whether rounding *v* to the nearest whole number will give 218 ounces (rounding down) or 219 ounces (rounding up). Specifically, we need to know whether √8192 is bigger or smaller than 90.5, so we calculate 90.5² = (90+0.5)² = 8100 + 2×0.5×90 + 0.25 = 8190.25, from which we conclude that the volume of the cooler rounds up to the nearest ounce, yielding **219** ounces.

**One Liner**. Coach Newton answers the one-liner with one picture:

There are 12 one-by-one squares, 6 two-by-two quares, and 2 three-by-three squares for a total of 20 squares with only **9** line segments.

## Recent Weeks

**Week 83**: Sour Lemons & One Liner, solutions to Curious Clock & Quadratic Triple

**Week 82**: Curious Clock & Quadratic Triple, solutions to Poor Pascal & Rascal’s Grid

**Week 81**: Poor Pascal & Rascal’s Grid, solutions to Secret Remainder & Party Puzzle

**Week 80**: Secret Remainder & Party Puzzle, solutions to Wing Nuts & Seven Eleven

**Week 79**: Wing Nuts & Seven Eleven, solutions to Polychromatic Pigs & Guest Gifts

Links to all of the puzzles and solutions are on the Complete Varsity Math page.

**Come back next week** for answers and more puzzles.