### 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?

## 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

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

Come back next week for answers and more puzzles.