(I) Assuming a lifespan of 80 years, in what years of the 20th and 21st centuries (1900-1999), (2000-2099) would you have to be born to have the maximum number of prime birthdays in a year whose sum of the digits is a prime number?
(II) In what years of the same time spans would you have to be born to have the minimum number of prime birthdays in a year whose sum of the digits is a prime number?
(For example, the sum of the digits of the year 1967 is 23, which is a prime number. People born in 1954 were 13 in 1967, which is also a prime number.)
Note: For the purposes of the problem, assume that that people born on Feb. 29 in a leap year still celebrate their birthdays each following year.
10 min=9999
20 for yr0 = 1900 to 2099
30 age=2:prCt=0
40 while age < 80
50 yr=yr0+age
60 sod=0:y=cutspc(str(yr))
70 for i=1 to len(y)
80 sod=sod+val(mid(y,i,1))
90 next
100 if prmdiv(sod)=sod and sod>1 then inc prCt
110 age=nxtprm(age)
120 wend
130 if prCt>max then max=prCt:maxYr=yr0
140 if prCt<min then min=prCt:minYr=yr0
150 next
160 print minYr;min
170 print maxYr;max
180 print
220 for yr0 = 1900 to 2099
230 age=2:prCt=0
240 while age < 80
250 yr=yr0+age
260 sod=0:y=cutspc(str(yr))
270 for i=1 to len(y)
280 sod=sod+val(mid(y,i,1))
290 next
300 if prmdiv(sod)=sod and sod>1 then inc prCt
310 age=nxtprm(age)
320 wend
330 if prCt=max or prCt=min then
340 :age=2 : print yr0,
350 :while age < 80
360 : yr=yr0+age
370 : sod=0:y=cutspc(str(yr))
380 : for i=1 to len(y)
390 : sod=sod+val(mid(y,i,1))
400 : next
410 : if prmdiv(sod)=sod and sod>1 then print age;yr;sod,:endif
420 : age=nxtprm(age)
430 :wend
440 :print:print
450 next
finds
1937 3
2061 14
meaning the minimum number is 3 and the maximum is 14, with these first occurring in 1937 and 2061 respectively.
Then, each of these numbers is shown to have two more occurrences within the time period:
1937 17 1954 19 53 1990 19 73 2010 3
1946 3 1949 23 17 1963 19 59 2005 7
1955 3 1958 23 17 1972 19 59 2014 7
2061 2 2063 11 11 2072 11 13 2074 13 17 2078 17
29 2090 11 31 2092 13 37 2098 19 41 2102 5
43 2104 7 47 2108 11 59 2120 5 61 2122 7
67 2128 13 79 2140 7
2067 2 2069 17 5 2072 11 7 2074 13 11 2078 17
23 2090 11 29 2096 17 31 2098 19 37 2104 7
41 2108 11 53 2120 5 59 2126 11 61 2128 13
73 2140 7 79 2146 13
2085 2 2087 17 5 2090 11 7 2092 13 11 2096 17
13 2098 19 17 2102 5 19 2104 7 23 2108 11
37 2122 7 41 2126 11 43 2128 13 59 2144 11
61 2146 13 79 2164 13
where, in each grouping, the year of birth is followed by subgroups of age, year and sum of digits of the year.
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Posted by Charlie
on 2011-07-14 13:16:36 |
computer solution
