• #### Simplify these scores for Jill and Debby

From James Dow Allen@21:1/5 to All on Thu Apr 16 00:20:54 2020
Jill and Debby agree to play a 6-event contest. There will be six events;
each event has one winner and is worth a certain number of points.
Here are the events:
F (Floor Show) 29 points
P (Pencil) 20 points
M (Mishmash) 16 points
N (No-Car) 15 points
W (Whiskey) 10 points
H (Hamper) 4 points
Jill wins if she gets 42 points or more.
Otherwise Debby gets 53 points or more and wins.

Neither Jill nor Debby is fond of arithmetic and each finds it tedious
to calculate which set of events they need to win. Can you help them?

Replace the point scores for the six events (along with the players'
point targets) with smaller positive integers that leave the players'
goals (the list of event subsets they need to win) unchanged.

Dehyphenatingly,
James-dow-allen at G-mail

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• From James Dow Allen@21:1/5 to James Dow Allen on Sat Apr 18 01:02:11 2020
I'm afraid my puzzle was too boring. :-(
I'll go ahead and post my intended follow-up now.

The puzzle (repeated below) has a solution of the form
P' = [P/r]
where P is the original point score, P' the maximally simplified score,
r some real number, and [_] takes the nearest integer.
New Puzzle: Find a different set of scores with a solution not of this form, nor of the form P' = min(A, max (B, [P/r])

James

On Thursday, April 16, 2020 at 2:20:57 PM UTC+7, James Dow Allen wrote:
Jill and Debby agree to play a 6-event contest. There will be six events; each event has one winner and is worth a certain number of points.
Here are the events:
F (Floor Show) 29 points
P (Pencil) 20 points
M (Mishmash) 16 points
N (No-Car) 15 points
W (Whiskey) 10 points
H (Hamper) 4 points
Jill wins if she gets 42 points or more.
Otherwise Debby gets 53 points or more and wins.

Neither Jill nor Debby is fond of arithmetic and each finds it tedious
to calculate which set of events they need to win. Can you help them?

Replace the point scores for the six events (along with the players'
point targets) with smaller positive integers that leave the players'
goals (the list of event subsets they need to win) unchanged.

Dehyphenatingly,
James-dow-allen at G-mail

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• From Edward Murphy@21:1/5 to James Dow Allen on Sun Apr 19 11:04:06 2020
On 4/16/2020 12:20 AM, James Dow Allen wrote:

Jill and Debby agree to play a 6-event contest. There will be six events; each event has one winner and is worth a certain number of points.
Here are the events:
F (Floor Show) 29 points
P (Pencil) 20 points
M (Mishmash) 16 points
N (No-Car) 15 points
W (Whiskey) 10 points
H (Hamper) 4 points
Jill wins if she gets 42 points or more.
Otherwise Debby gets 53 points or more and wins.

Neither Jill nor Debby is fond of arithmetic and each finds it tedious
to calculate which set of events they need to win. Can you help them?

Replace the point scores for the six events (along with the players'
point targets) with smaller positive integers that leave the players'
goals (the list of event subsets they need to win) unchanged.

Dehyphenatingly,
James-dow-allen at G-mail

For completeness, here's a (hopefully correct) solution to the original:

Debby has a higher target, so her options are fewer.

If she wins F, then these suffice:

* F + P + at least one of MNWH
* F + M + at least one of NW
* F + N + W

If she doesn't win F, then she can miss at most 12 of 65, so she must
win P + M + N + at least one of WH.

Let's try halving everything (rounding down):

F 14
P 10
M 8
N 7
W 5
H 2

* F + P + at least one of MNWH = 26+
* F + M + at least one of NW = 27+
* F + N + W = 26
* P + M + N + at least one of WH = 27+

* F + P = 24
* F + M + H = 24
* F + N + H = 23
* F + W + H = 21
* P + M + N = 25
* M + N + W + H = 22

So they can use these values and Debby must get 26 or more.

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