Question

Problem B

Years and Years of Threepeats'
Find all years between 1000 BCE and 2020 CE, inclusive, which have exactly
three digits the same. For example, the years 1011 and 1222 each have exactly
three digits the same, but 1111 and 1123 do not.

Answer 1

The years are:

• 1000 BCE, 999 BCE, 888 BCE, 777 BCE, 666 BCE, 555BCE, 444 BCE. 333 BCE, 222 BCE, 111 BCE

• 111 CE, 222 CE, 333 CE, 444 CE, 555 CE, 666 CE, 777 CE, 888 CE, 999 CE, 1000CE, 1011 CE, 1101 CE, 1110 CE,1222CE, 1333CE, 1444CE, 1555 CE, 1666 CE, 1777CE, 1888 CE, 1999CE, and 2000 CE

Step-by-step explanation:

1. Years BC:

a) Years with four digits:

The first number with 3 equal digits is 1000. After that the years go decreasing: 999, 998, 997, ...

b) Years with three digits:

From 999 to 111, the numbers have three digits, thus the only that are solutions ara 999, 888, 777, 666, 555, 444, 333, 222, and 111: 9 numbers

After that the years have two digits, thus no solutions, with two digits.

Hence, we count 10 different years.

2. Years CE

a) Years with three digits:

• 111, 222, 333, 444, 555, 666, 777, 888, 999: 9 years

b) Years with four digits

i) Starting with 1:

• With three 0: 1000: 1 year

• With three 1: 1011, 1101, 1110: 3 years

• With three digits different to 1: 1222, 1333, 1444, 1555, 1666, 1777, 1888, 1999: 8 years

ii) Starting with 2:

• With three 0: 2000: 1 year

The next one with three equal digits is 2111 and it is after 2020 CE.

Therefore, 9 + 1 + 3 + 8 + 1 = 22 years starting with 2.

3. Total

10 years BC and 22 years CE have exactly three digits the same: 10 + 22 = 32.