Varsity Math, Week 114

Varsity Math 114

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?

Gifts

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.