10. What does the year more often start with: Saturday or Sunday?



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10. Note that if among 28 consecutive years every fourth year is a leap year (i.e., there is no year whose number is divisible by 100 but not by 400), then these 28 years contain a whole number of weeks. Since January 1, 1901, was a Tuesday, January 1, 1929, will also be a Tuesday. It can be calculated that during these 28 years, New Year's Day will start on Saturdays and Sundays an equal number of times. Note that 400 years also contain a whole number of weeks (since the number of additional days per year is 497: 3 days are lost due to the "lost" leap years). Therefore, the period of our calendar is not 28 but 400 years.

|  | Days on which New Year starts from 1901 |  |  |  |  |
| :---: | :---: | :---: | :---: | :---: | :---: |
| Monday | 1906 | 1912 | 1928 |  |  |
| Tuesday | 1901 | 1907 | 1918 | 1923 |  |
| Wednesday | 1902 | 1908 | 1913 | 1919 | $\mathbf{1 9 2 9}^{2}$ |
| Thursday | 1903 | 1914 | 1920 | 1925 |  |
| Friday | 1904 | 1909 | 1915 | 1926 |  |
| Saturday | 1910 | 1916 | 1921 | 1927 |  |
| Sunday | 1905 | 1911 | 1922 | 1928 |  |

The distribution of the days of the week will be the same for the years 1929-1956, 1957-1984, and so on until 2096 (since the year 2000 is a leap year). Thus, January 1, 2097, will be a Tuesday. Then, January 1, 2098, will be a Wednesday, January 1, 2099, will be a Thursday, January 1, 2100, will be a Friday, and January 1, 2101, will be a Saturday.

Similarly, in the next 28-year periods (2101-2128, 2129-2156, and 2157-2184), New Year's Day will start an equal number of times on each day of the week, and January 1, 2129, 2157, and 2185, will be a Saturday. Further:

|  | Days on which New Year starts from 2185 to 2200 |  |  |  |
| :---: | :---: | :---: | :---: | :---: |
| Monday |  | 2187 |  | 2198 |
| Tuesday |  | 2188 | 2193 | 2199 |
| Wednesday |  |  | 2194 | 2200 |
| Thursday |  | 2189 | 2195 | 2201 |
| Friday |  | 2190 | 2196 |  |
| Saturday | 2185 | 2191 |  |  |
| Sunday | 2186 | 2192 | 2197 |  |

Similarly, over the next 84 years (2201 to 2284), New Year's Day will also start an equal number of times on all days of the week, and January 1, 2185, will be a Thursday.

|  | Days on which New Year starts from |  |  |  |
| :---: | :---: | :---: | :---: | :---: |
| 2285 to 2300 |  |  |  |  |
| Monday |  |  | 2294 | $2300-$ cycle |
|  |  | 2289 | 2295 |  |
| Tuesday |  | 2290 | 2296 |  |
| Wednesday | 2285 | 2291 |  |  |
| Thursday | 2286 | 2292 | 2297 |  |
| Friday | 2286 |  |  |  |
| Saturday | 2287 |  | 2298 |  |
| Sunday | 2288 | 2293 | 2299 |  |

Thus, during the years 1901-2096, 2101-2184, and 2201-2284, New Year's Day will start an equal number of times on all days of the week, while the remaining years in the period 1901-2300 will give 6 Sundays and 4 Saturdays. Therefore, New Year's Day starts on a Sunday more often than on a Saturday.