**Problem: **What was the day of the week on 16th July, 1776?

**Solution:**

16th July, 1776 = (1775 years + Period from 1.1.1776 to 16.7.1776)

**Counting of odd days:**

Number of odd days in 1600 years = 0

Number of odd days in 100 years = 5

75 years = 18 leap years + 57 ordinary years

= (18 × 2 + 57 × 1) odd days = 93 odd days

= (13 weeks + 2 days) ≡ 2 odd days

∴ 1775 years have

= (0 + 5 + 2) odd days = 7 odd days ≡ 0 odd day.

Jan. Feb. March April May June July (31 + 29 + 31 + 30 + 31 + 30 + 16) = 198 days

198 days = (28 weeks + 2 days) ≡ 2 odd days.

∴ Total number of odd days = (0 + 2) = 2.

Hence, the required day is Tuesday.

**Problem:** What was the day of the week on 15th August, 1947?

**Solution:**

15th August, 1947 = (1946 years + Period from 1.1.1947 to 15.8.1947)

Odd days in 1600 years = 0

Odd days in 300 years = (5 × 3) = 15 ≡ 1

46 years = (11 leap years + 35 ordinary years)

= (11 × 2 + 35 × 1) odd days = 57 odd days

= (8 weeks + 1 day) ≡ 1 odd day.

∴ Odd days in 1946 years = (0 + 1+ 1) = 2.

Jan. Feb. March April May June July Aug

(31 + 28 31 + 30 + 31 + 30 + 31 + 15) = 227 days

227 days = (32 weeks + 3 days) ≡ 3 odd days.

Total number of odd days = (2 + 3) = 5.

Hence, the required day is Friday.

**Problem:** What was the day of the week on 4th June, 2002?

**Solution:**

4th June, 2002 = (2001 years + Period from 1.1.2002 to 4.6.2002)

Odd days in 1600 years = 0

Odd days in 400 years = 0

Odd days in 1 ordinary year = 1

Odd days in 2001 years = (0 + 0 + 1) = 1

Jan. Feb. March April May June

= 155 days

(31 + 28 + 31 + 30 + 31 + 4)

= 22 weeks + 1 day ≡ 1 odd day

Total number of odd days = (1 + 1) = 2 ∴ Required day is Tuesday.

**Problem:** On what dates of March 2005 did Friday fall?

**Solution:**

First, we find the day on 1.3.2005

1.3.2005 = (2004 years + Period from 1.1.2005 to 1.3.2005)

Odd days in 1600 years = 0 Odd days in 400 years = 0

4 years = (1 leap year + 3 ordinary years)

= (1 × 2 + 3 × 1) odd days = 5 odd days

Jan. Feb. March

(31 + 28 + 11)

= 60 days = (8 weeks + 4 days) ≡ 4 odd days.

Total number of odd days = (0 + 0 + 5 + 4) = 9 ≡ 2 odd days

∴ 1.3.2005 was Tuesday. So, Friday lies on 4.3.2005 Hence, Friday lies on 4th, 11th, 18th and 25th of March, 2005

**Problem:** Prove that the calendar for the year 2003 will serve for the year 2014.

**Solution:**

We must have the same day on 1.1.2003 and 1.1.2014.

So, the number of odd days between 31.12.2002 and 31.12.2013 must be 0.

This period has 3 leap years and 8 ordinary years.

Number of odd days = (3 × 2 + 8 × 1) = 14 ≡ 0 odd day

∴ Calendar for the year 2003 will serve for the year 2014.

**Problem:** January 1, 2007 was Monday. What day of the week lies on Jan. 1, 2008?

**Solution:**

The year 2007 is an ordinary year. So, it has 1 odd day.

1st day of the year 2007 was Monday.

1st day of the year 2008 will be 1 day beyond Monday. Hence, it will be on Tuesday.

**Problem:** January 1, 2008 is Tuesday. What day of the week lies on Jan. 1, 2009?

**Solution:**

The year 2008 is a leap year. So, it has 2 odd days.

1st day of the year 2008 is Tuesday (Given)

So, 1st day of the year 2009 is 2 days beyond Tuesday.

Hence, it will be Thursday.

**Problem:** On 8th Dec, 2007 Saturday falls. What day of the week was it on 8th Dec. 2006?

**Solution:**

The year 2006 is an ordinary year. So, it has 1 odd day.

So, the day on 8th Dec, 2007 will be 1 day beyond the

day on 8th Dec, 2006.

But, 8th Dec, 2007 is Saturday.

∴ 8th Dec, 2006 is Friday.

**Problem:** On 6th March, 2005 Monday falls. What was the day of the week on 6th March, 2004?

**Solution:**

The year 2004 is a leap year. So, it has 2 odd days.

∴ The day on 6th March, 2005 will be 2 days beyond the day on 6th March, 2004.

But, 6th March, 2005 is Monday.

∴ 6th March, 2004 is Saturday.

**Problem: **The calendar for the year 2007 will be the same for the year: (a) 2014 (b) 2016 (c) 2017 (d) 2018

Count the number of odd days from the year 2007 onwards to

get the sum equal to 0 odd day.

Year | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 |

Odd day | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 |

Sum = 14 odd days ≡ 0 odd day.

∴ The calendar for the year 2018 will be the same as for the year 2007.

**Problem:** On what dates of April, 2001 did Wednesday fall?
(a) 1st, 8th, 15th, 22nd, 29th
(b) 2nd, 9th, 16th, 23rd, 30th
(c) 3rd, 10th, 17th, 24th
(d) 4th, 11th, 18th, 25th

**Solution:**

We shall find the day on 1st April, 2001.

1st April, 2001 = (2000 years + Period from 1.1.2001 to

1.4.2001)

Odd days in 1600 years = 0

Odd days in 400 years = 0

Jan. Feb. March April

= 91 days ≡ 0

odd days. (31 + 28 + 31 + 1)

**Problem:** What was the day of the week on 17th June, 1998?
(a) Monday (b) Tuesday
(c) Wednesday (d) Thursday

**Solution:**

Wednesday

**Problem:** What was the day of the week on 28th May, 2006?
(a) Thursday (b) Friday
(c) Saturday (d) Sunday

**Solution:**

Sunday

**Problem:** What will be the day of the week on 15th August, 2010?
(a) Sunday (b) Monday
(c) Tuesday (d) Friday

**Solution:**

Sunday

**Problem:** Today is Monday. After 61 days, it will be
(a) Wednesday (b) Saturday
(c) Tuesday (d) Thursday

**Solution:**

Saturday

**Problem:** The last day of a century cannot be
(a) Monday (b) Wednesday
(c) Tuesday (d) Friday

**Solution:**

Tuesday

**Problem:** Which of the following is not a leap year?
(a) 700 (b) 800
(c) 1200 (d) 2000

**Solution:**

700

**Problem:** How many days are there in x weeks x days?
(a) 7x2 (b) 8x
(c) 14x (d) 7

**Solution:**

*8x*

**Problem:** It was Sunday on Jan 1, 2006. What was the day of the week on Jan 1, 2010?
(a) Sunday (b) Saturday
(c) Friday (d) Wednesday

**Solution:**

Friday

**Problem:**On 8th Feb, 2005 it was Tuesday. What was the day of the week on 8th Feb, 2004?
(a) Tuesday (b) Monday
(c) Sunday (d) Wednesday

**Solution:**

Sunday

**Problem:** For a certain month, the dates of three of the Sundays are even numbers. Then, the 15th of the that month falls on a [SSC—CGL (Tier I) Exam, 2012]
(a) Thursday (b) Friday
(c) Saturday (d) Sunday

**Solution:**

The dates of three of the Sundays are even number is

2, 9, 16, 23, 30.

So, on 16th of that month = Sunday, 15th of that month falls on a Saturday

**Problem:** What was the day of the week on 15 August, 1947? [DMRC— Customer Relationship Assistant (CRA) Exam, 2016]
(a) Saturday (b) Friday
(c) Thursday (d) Wednesday

**Solution:**

15 August 1947 means 1946 complete years + first 7

months up to July 1947 + 15 days of August 1947

1600 years have 0 odd days.

300 years have 1 odd day

46 years have 11 leap years and 35 ordinary years

= (11 × 2) + (35 × 1)

= 22 + 35 = 57 odd days

= 8 × 7 + 1 odd days

= 8 weeks + 1 odd day

Up to 1946 there are 1 + 1 = 2 odd days

January 1947 ⇒ 3 odd days

February 1947 ⇒ 0 odd days

(1947 is a normal year)

March 1947 ⇒ 3 odd days

April 1947 ⇒ 2 odd days

May 1947 ⇒ 3 odd days

June 1947 ⇒ 2 odd days

July 1947 ⇒ 3 odd days

Up to 15 August ⇒ 15 odd days

Total number of odd days up to 15 August 1947

= 2 + 3 + 0 + 3 + 2 + 3 + 2 + 3 + 15 = 33 odd days.

Hence, 15th August 1947 was Friday.

**Problem:** The calendar for the year 2009 will be the same as that of the year [DMRC— Customer Relationship Assistant (CRA) Exam, 2016]
(a) 2013 (b) 2014
(c) 2015 (d) 2014

**Solution:**

2008 was a leap year.

A leap year has two odd days.

Suppose the year 2008 starts with a Monday.

Then the first day of 2009 was Wednesday.

Now,

First day of 2010 ⇒ Thursday

First day of 2011 ⇒ Friday

First day of 2012 ⇒ Saturday

Because 2012 was a leap year

First day of 2013 ⇒ Monday

First day of 2014 ⇒ Tuesday

First day of 2015 ⇒ Wednesday

Thus, the calendar for the year 2009 was the same as that of the year 2015.