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1.	Prove that any date in March of a year is the same day of the week corresponding date in November that year.
Answer» We will show that the number of odd days between last day of February and last day of October is zero. March April May June July Aug. Sept. Oct. 31 + 30 + 31 + 30 + 31 + 31 + 30 + 31 = 241 days = 35 weeks = 0 odd day. Number of odd days during this period = 0. Thus, 1st March of an year will be the same day as 1st November of that year. Hence, the result follows.

Prove that any date in March of a year is the same day of the week corresponding date in November that year.

Answer»

We will show that the number of odd days between last day of February and last day of October is zero.

March April May June July Aug. Sept. Oct.

31 + 30 + 31 + 30 + 31 + 31 + 30 + 31 = 241 days = 35 weeks = 0 odd day.

Number of odd days during this period = 0.

Thus, 1st March of an year will be the same day as 1st November of that year. Hence, the result follows.

Discussion

2.	What was the day of the week on 16th April, 2000 ?
Answer» 16th April, 2000 = (1999 years + Period from 1st Jan., 2000 to 16th A) Counting of odd days: 1600 years have 0 odd day. 300 years have 1 odd day. 99 years = (24 leap years + 75 ordinary years) = [(24 x 2) + (75 x 1)] odd days = 123 odd days = (17 weeks + 4 days) = 4 odd days. Jan. Feb. March April 31 + 29 + 31 + 16 = 107 days = (15 weeks + 2 days) = 2 odd, Total number of odd days = (0 + 1 + 4 + 2) odd days = 7 odd days = 0 odd day. Hence, the required day was 'Sunday'.

What was the day of the week on 16th April, 2000 ?

Answer»

16th April, 2000 = (1999 years + Period from 1st Jan., 2000 to 16th A)

Counting of odd days:

1600 years have 0 odd day. 300 years have 1 odd day.

99 years = (24 leap years + 75 ordinary years)

= [(24 x 2) + (75 x 1)] odd days = 123 odd days

= (17 weeks + 4 days) = 4 odd days.

Jan. Feb. March April

31 + 29 + 31 + 16 = 107 days = (15 weeks + 2 days) = 2 odd,

Total number of odd days = (0 + 1 + 4 + 2) odd days = 7 odd days = 0 odd day. Hence, the required day was 'Sunday'.

Discussion

3.	What was the day of the week on 16th August, 1947?
Answer» 15th August, 1947 = (1946 years + Period from 1st Jan., 1947 to 15th Counting of odd days: 1600 years have 0 odd day. 300 years have 1 odd day. 47 years = (11 leap years + 36 ordinary years) = [(11 x 2) + (36 x 1») odd days = 58 odd days = 2 odd days. Jan. Feb. March April May June July Aug. 31 + 28 + 31 + 30 + 31 + 30 + 31 + 15 = 227 days = (32 weeks + 3 days) = 3, Total number of odd days = (0 + 1 + 2 + 3) odd days = 6 odd days. Hence, the required day was 'Saturday'.

What was the day of the week on 16th August, 1947?

Answer»

15th August, 1947 = (1946 years + Period from 1st Jan., 1947 to 15th

Counting of odd days:

1600 years have 0 odd day. 300 years have 1 odd day.

47 years = (11 leap years + 36 ordinary years)

= [(11 x 2) + (36 x 1») odd days = 58 odd days = 2 odd days.

Jan. Feb. March April May June July Aug.

31 + 28 + 31 + 30 + 31 + 30 + 31 + 15

= 227 days = (32 weeks + 3 days) = 3,

Total number of odd days = (0 + 1 + 2 + 3) odd days = 6 odd days.

Hence, the required day was 'Saturday'.

Discussion

4.	On what dates of July 2004 did Monday fall?
Answer» Let us find the day on 1st July, 2004. 2000 years have 0 odd day. 3 ordinary years have 3 odd days. Jan. Feb. March April May June July 31 + 29 + 31 + 30 + 31 + 30 + 1 = 183 days = (26 weeks + 1 day) = 1 t . Total number of odd days = (0 + 3 + 1) odd days = 4 odd days. :. 1st July 2004 was 'Thursday',-,- Thus, 1st Monday in July 2004 as on 5th July. Hence, during July 2004, Monday fell on 5th, 12th, 19th and 26th.

On what dates of July 2004 did Monday fall?

Answer»

Let us find the day on 1st July, 2004.

2000 years have 0 odd day. 3 ordinary years have 3 odd days.

Jan. Feb. March April May June July

31 + 29 + 31 + 30 + 31 + 30 + 1

= 183 days = (26 weeks + 1 day) = 1 t .

Total number of odd days = (0 + 3 + 1) odd days = 4 odd days.

:. 1st July 2004 was 'Thursday',-,-

Thus, 1st Monday in July 2004 as on 5th July.

Hence, during July 2004, Monday fell on 5th, 12th, 19th and 26th.

Discussion