The proprietor of Ye Olde Puzzle Shoppe is a bit eccentric.   Recently, he baffled some of his would-be customers by putting this sign in the shop's window:

When we're open, our hours are from noon to 6:00 PM.
Our weekly schedule is as follows:

1. We're open for at least three consecutive days.

2. All the days that we're open start with different letters of the alphabet.

3. The total number of letters in the names of the days that we're open is odd.

4. Only two of the days that we're open have names of the same length.   All our other open days have names of different lengths.

5. We're either open on both Monday and Wednesday or closed on both Monday and Wednesday.


So which days of the week is the shop open?


Solution:


The shop is open each Monday, Wednesday, Thursday, and Friday.

First, determine the number of letters in each name:
SUNDAY -- 6
MONDAY -- 6
TUESDAY -- 7
WEDNESDAY -- 9
THURSDAY -- 8
FRIDAY -- 6
SATURDAY -- 8

Since Tuesday and Wednesday are the only days with an odd number of letters in their names, the shop must be open one of these days but not both (clue 3).

The shop cannot be open both Tuesday and Thursday, nor both Saturday and Sunday (clue 2).

With these restrictions in mind, the only possible three-day sequences that the shop can be open are:
Sunday-Monday-Tuesday
Wednesday-Thursday-Friday
Thursday-Friday-Saturday

We can eliminate the Sunday-Monday-Tuesday sequence because clue 5 requires the shop to be open on Wednesday but then that violates clue 3.

We can eliminate the Thursday-Friday-Saturday sequence because it leads to contradiction with clue 3.
    If the shop is open on Wednesday, it must be also be open on Monday, but then
    Monday and Friday are of the same length and Thursday and Saturday are the same length which
    violates clue 4.
    Since the shop can't be open on Wednesday, it must be open on Tuesday (clue 3), but that violates clue 2.

That leaves Wednesday-Thursday-Friday as the only three-day sequence that works.
    Clue 5 then requires the shop to be open on Monday.
    Monday, Wednesday, Thursday, and Friday satisfy all the clues.
    The shop cannot be open on Sunday or Saturday because that contradicts clue 4.
    The shop cannot be open on Tuesday because that contradicts clue 3.


Correctly solved by:

1. K. Sengupta Calcutta, India
2. Kamal Lohia Holy Angel School,
Hisar, Haryana, India
3. Colin (Yowie) Bowey Beechworth, Victoria, Australia
4. Davit Banana Istanbul, Turkey
5. Ivy Joseph Pune, Maharashtra, India
6. Kelly Stubblefield Mobile, Alabama, USA