(A) The fact that 14 years is asked for makes one think that what is sought is years of all 14 types--ordinary vs leap, and beginning on any of the 7 days of the week. But 14 years in a row will not produce this complete set as there are only three or four leap years in such a consecutive set, which thus cannot be all seven beginning days of the week.
So for part (A) I made up this table, showing for each of the 14 types of year, the appropriate months with five Thursdays or Sundays as appropriate for the parity of the month number:
Year Starting List of day of week with month number
type On
ordinary Sunday Th 3; Su 4; Su 10; Th 11; Su 12;
ordinary Monday Th 3; Su 4; Th 5; Th 11; Su 12;
ordinary Tuesday Th 1; Th 5; Su 6; Su 12;
ordinary Wednesday Th 1; Th 5; Su 6; Th 7; Su 8;
ordinary Thursday Th 1; Th 7; Su 8;
ordinary Friday Th 7; Su 8; Th 9; Su 10;
ordinary Saturday Th 3; Th 9; Su 10;
leap Sunday Th 3; Su 4; Th 5; Th 11; Su 12;
leap Monday Th 5; Su 6; Su 12;
leap Tuesday Th 1; Th 5; Su 6; Th 7; Su 8;
leap Wednesday Th 1; Th 7; Su 8;
leap Thursday Th 1; Su 2; Th 7; Su 8; Th 9; Su 10;
leap Friday Th 3; Th 9; Su 10;
leap Saturday Th 3; Su 4; Su 10; Th 11; Su 12;
The program builds up a yearGrid, showing the number of club meetings Arne can attend in the calendar year:
ordinary leap
Sunday 17 17
Monday 17 15
Tuesday 16 17
Wednesday 17 15
Thursday 15 18
Friday 16 15
Saturday 15 17
There are a minimum of 15 and maximum of 18 in a given year.
The above was produced by the first part of the program:
clearvars, clc
ordinary=[31 28 31 30 31 30 31 31 30 31 30 31];
leap=[31 29 31 30 31 30 31 31 30 31 30 31];
dNames=["Sunday","Monday","Tuesday","Wednesday","Thursday","Friday","Saturday"];
for typeY=["ordinary","leap"]
for start=[1:7]
totInTypeStart=0;
eval(['lengths=' char(typeY) ';']);
sunday1=9-start;
if sunday1>7
sunday1=sunday1-7;
end
fprintf('%8s %10s',typeY,dNames(start));
for mo=1:12
thursday1=sunday1+4;
if thursday1>7
thursday1=thursday1-7;
end
if mod(mo,2)==1
thursday5=thursday1+28;
if thursday5<=lengths(mo)
fprintf(' Th %2d;',mo);
totInTypeStart=totInTypeStart+1;
end
totInTypeStart=totInTypeStart+1;
else
sunday5=sunday1+28;
if sunday5<=lengths(mo)
fprintf(' Su %2d;',mo);
totInTypeStart=totInTypeStart+1;
end
totInTypeStart=totInTypeStart+1;
end
sunday1=sunday1+28-lengths(mo);
if sunday1<1
sunday1=sunday1+7;
end
end
fprintf('\n');
yearGrid(start,1+(typeY=="leap"))=totInTypeStart;
end
end
January 1, 2023, will be a Sunday, and that year is the year before a leap year. So the second part of the same program uses the grid built in the first part:
start=1;
for yr=2023:2036
fprintf('%4d ',yr);
if mod(yr,4)==0
fprintf('leap %12s',dNames(start));
fprintf(' %2d',yearGrid(start,2));
else
fprintf('ordinary %12s',dNames(start))
fprintf(' %2d',yearGrid(start,1));
end
fprintf('\n')
start=start+1; % ordinarily new year starts with next day of week
if mod(yr,4)==0 % mod valid only between 1901 and 2099
start=start+1; % leap year leaps over that day to the next
end
if start>7
start=start-7;
end
end
producing
part (B)
year type beginning number of
on meetings
2023 ordinary Sunday 17
2024 leap Monday 15
2025 ordinary Wednesday 17
2026 ordinary Thursday 15
2027 ordinary Friday 16
2028 leap Saturday 17
2029 ordinary Monday 17
2030 ordinary Tuesday 16
2031 ordinary Wednesday 17
2032 leap Thursday 18
2033 ordinary Saturday 15
2034 ordinary Sunday 17
2035 ordinary Monday 17
2036 leap Tuesday 17
The maximum number of meetings, 18, is indeed achieved in 2032, as this 14-year period does include a leap year beginning on a Thursday.
The minimum, 15, occurs in 2024, 2026, and 2033.
A complete cycle, between the years 1901 and 2099 contains 28 years: one of each leap year beginning day and four of each ordinary year with any given starting day of the week.
From 2033 through 2099, it's
2023 ordinary Sunday 17
2024 leap Monday 15
2025 ordinary Wednesday 17
2026 ordinary Thursday 15
2027 ordinary Friday 16
2028 leap Saturday 17
2029 ordinary Monday 17
2030 ordinary Tuesday 16
2031 ordinary Wednesday 17
2032 leap Thursday 18
2033 ordinary Saturday 15
2034 ordinary Sunday 17
2035 ordinary Monday 17
2036 leap Tuesday 17
2037 ordinary Thursday 15
2038 ordinary Friday 16
2039 ordinary Saturday 15
2040 leap Sunday 17
2041 ordinary Tuesday 16
2042 ordinary Wednesday 17
2043 ordinary Thursday 15
2044 leap Friday 15
2045 ordinary Sunday 17
2046 ordinary Monday 17
2047 ordinary Tuesday 16
2048 leap Wednesday 15
2049 ordinary Friday 16
2050 ordinary Saturday 15 ________
2051 ordinary Sunday 17
2052 leap Monday 15
2053 ordinary Wednesday 17
2054 ordinary Thursday 15
2055 ordinary Friday 16
2056 leap Saturday 17
2057 ordinary Monday 17
2058 ordinary Tuesday 16
2059 ordinary Wednesday 17
2060 leap Thursday 18
2061 ordinary Saturday 15
2062 ordinary Sunday 17
2063 ordinary Monday 17
2064 leap Tuesday 17
2065 ordinary Thursday 15
2066 ordinary Friday 16
2067 ordinary Saturday 15
2068 leap Sunday 17
2069 ordinary Tuesday 16
2070 ordinary Wednesday 17
2071 ordinary Thursday 15
2072 leap Friday 15
2073 ordinary Sunday 17
2074 ordinary Monday 17
2075 ordinary Tuesday 16
2076 leap Wednesday 15
2077 ordinary Friday 16
2078 ordinary Saturday 15______________
2079 ordinary Sunday 17
2080 leap Monday 15
2081 ordinary Wednesday 17
2082 ordinary Thursday 15
2083 ordinary Friday 16
2084 leap Saturday 17
2085 ordinary Monday 17
2086 ordinary Tuesday 16
2087 ordinary Wednesday 17
2088 leap Thursday 18
2089 ordinary Saturday 15
2090 ordinary Sunday 17
2091 ordinary Monday 17
2092 leap Tuesday 17
2093 ordinary Thursday 15
2094 ordinary Friday 16
2095 ordinary Saturday 15
2096 leap Sunday 17
2097 ordinary Tuesday 16
2098 ordinary Wednesday 17
2099 ordinary Thursday 15
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Posted by Charlie
on 2022-05-17 12:27:15 |
solution as I think intended
