### Primes and Products

Pete and Dick share a birthday in early November but are six years apart in age.

Dick tells the team: “Pete’s age is now a prime, and my age is now the product of two odd primes.”

Pete adds, “This last happened 12 years ago.”

*How old is Dick?*

### Two Pints of Cider

Team member Janice recently visited the U.K. and poses this puzzle to her teammates: You have three containers that can hold exactly 15, 10, and 6 pints. The 15-pint container starts full of cider. You want to measure out exactly 2 pints of cider, drink it all, and end with an empty 15-pint container and 8 and 5 pints of cider in the other two containers.

*What transfers should you make to accomplish this?*

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## Solutions to week 113

In Prime Presents 1 there were six gifts with a bill of $2 + $3 + $5 + $41 + $67 + $89 = $207. In Prime Presents 2 there were six gifts with a bill of $2 + $3 + $5 + $67 + $89 + $401 = $567.

**Prime Presents 1 answer explained:** A bill of $2 + $3 + $5 + $41 + $67 + $89 gives a total $207. To use $8 we must use $83 or $89; to use $6 we must use $61 or $67 and to use $5 we must use $5, $53, or $59. This requires that one of the gifts was $5, otherwise, we would get a total greater than $207. To use $1, $4, and $6, we must use $41 and $67 or $47 and $61. The best way to use $2, $3, $8, and $9 is then $2 + $3 + $89. Note that using $41 + $67 or $47 + $61 gives an ambiguity in what the individual gifts could have cost.

**Prime Presents 2 answer explained:**

A bill of $2 + $3 + $5 + $67 + $89 + $401 gives a total $567. It’s tempting to use $103, $107 or $109 to get the $0 taken care of but for each of these the $4, $6, and $8 cannot all be used. The next primes using a $0 are $401 and $409. With $401 the only possibility for the $6 is in $67 and the rest fall into place. For $409 the only possibility for the $8 is in $89 and the rest fall into place as $2 + $5 + $7 + $61 + $83 + $409 = $567. Note that there is an ambiguity in what the individual gifts could have cost.

## Recent Weeks

**Week 113**: Prime Presents 1 & Prime Presents 2, solutions to Precise Prescription & Gift Dilemma

**Week 112**: Precise Prescription & Gift Dilemma, solutions to Pumpkin Crop & The Halloween Goblin

**Week 111**: Pumpkin Crop & The Halloween Goblin, solutions to Making 25 & Two Ways to 32

**Week 110**: Making 25 & Two Ways to 32, solutions to Commuters’ Dilemma & Construction Conundrum

**Week 109**: Commuters’ Dilemma & Construction Conundrum, solutions to Nine Coins & Five Questionable Coins

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

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