Pali (who is almost 15 years old) painted a beautiful calendar for herself at the beginning of the year, marking which day of the week each date falls on. Now she says: “I will use it up, because until I turn 50, I will still be able to use it for 5 more years.” Is Pali right?[^0]


[^0]:    ${ }^{1}$ It appeared in the December 1966 issue.
1. Pali made his calendar for 1966, which is an ordinary year, so it is obvious that it can be reused in any ordinary year where, for example, the beginning of January falls on the same day of the week as January 1, 1966, which is a Saturday; however, it cannot be used in other ordinary years.

In leap years, the calendar can never be used for the entire year, but only up to February 28, which aligns with the above condition, or from March 1 to the end of the year, provided that March 1 falls on the same day of the week as March 1, 1966. This condition is similarly met if the New Year's Day of the leap year falls on a Friday, which is one day earlier than January 1, 1966.

Accordingly, it is sufficient to examine the beginning of January until Pali turns 50, i.e., until 2001. This is facilitated by the following observation: the New Year's Day of a year following an ordinary year falls one day later in the week, and the New Year's Day of a year following a leap year falls two days later in the week, since an ordinary year consists of 52 full weeks and one day: \(365 = 52 \cdot 7 + 1\), and a leap year consists of 52 weeks and two days. According to this, we filled in the years in the rows of the days of the week in the table below, highlighting in bold those that meet the condition.

| New Year falls |  | year | in the | year |  |
| :---: | :---: | :---: | :---: | :---: | :---: |
| on Monday | 1968 | 1973,1979 |  | 1990,1996 | 2001 |
| on Tuesday |  | 1974 | 1980,1985 | 1991 |  |
| on Wednesday | 1969 | 1975 | 1986 | 1992, 1997 |  |
| on Thursday |  | 1970,1976 | 1981, 1987 | 1998 |  |
| on Friday |  | 1971 | \(1982, \mathbf{1 9 8 8 * *}\) | 1993, 1999 |  |
| on Saturday |  | \(1972^{*}, 1977\) | 1983 | 1994 | \(2000^{*}\) |
| on Sunday | 1967 | 1978 | 1984,1989 | 1995 |  |

According to this, Pali can use his calendar in 3 ordinary years, the first two months of 2 leap years, and the last 10 months of one leap year until he turns 50, totaling 4 years and 2 months; so he was not right. (His mistake can be explained by the following hasty "reconsideration": there are 35 years left, and \(1/7\) of this is five years.)

Jakál László (Sopron, Széchenyi I. g. I. o. t.)[^0]


[^0]:    ${ }^{1}$ Published in the December 1966 issue.