yeah!!! a century!
Level
2
Which of the following can't be the last dayof a century????
tuesday
friday
wednesday
monday
sunday
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1 solution
I've thought about it quite some and here's what i've come up with.
First a leap year is a year which has 1 day more than a normal year. A year is a leap year if: 1. it is divisable by 4 and not divisable by 100 or 2. it is divisable by 100 and divisable by 400
in a century there are 25 leap years if it ends with a leap year and there are 24 leap years if it does not
if there are 25 leap years in a century then the "shift" in days is ((365 100)+25) mod 7 = 6 if there are 24 leap years then the "shift" is ((365 100)+24) mod 7 = 5
this shift is the difference in days on the same date for instance if a day x is a monday the day x+5 would be a friday ( because friday is 5 days away from monday) this goes back as well so the day x-2 is saturday (which is 2 days back from monday) notice that x+7 is actually the same day x (so 7 days after a monday comes again a monday)
so now that we said all that stuff lets look at the 1st of jan on the first year of a century if the first day of a century is x the first day of the next century will be (according to the shift mentioned earlier): x-1 (which is the same as x+6) if the century ends with a leap year x-2 (which is the same as x+5) if the century doesn't end with a leap year
now lets take some starting year of a century which ends with a leap year for example 1901 (because the 20th century ends with 2000 and that is a leap year) ... 1901 starts with day x (first day of next cent. we substract 1) 2001 starts with day x-1 (first day of next cent we substract 2) 2101 starts with day x-3 (first day of next cent we substract 2) 2201 starts with day x-5 (first day of next cent we substract 2) 2301 starts with day x-7 (we substract 1 for the next...) ... this repeats itself int the past and future in 2301 we have the exact same situation as in 1901 as far as days go. so we see now that the days x-2 x-4 and x-6 can not start a century and knowing that 1901 started with a tuesday we know that x-2 = sunday x-4 = friday and x-6=wednesday can not start a century and the days preeceding sun fri and wed cannot end it and those are sat thur and tue.
So we conclude that tuesday thursday and saturday can not end a century.
I know i've made this freakin complicated but i believe its ok, dont hesitate to check mistakes are possible :P
P.S. Thursday can end 2099 but the officially last day of 21century is 31. dec 2100 and it is a friday