If you liked the last one…more 3, 5, 7
Did you like this puzzle? Then you might like to try this one that I found on a French math olympiad site:
Le grand mathématicien suisse Léonhart Euler est né à Bâle le 15 avril 1707. Il meurt plus d’un demi-siècle plus tard à Saint Pétersbourg en laissant une ouvre mathématique considérable. Dans une vie qui a duré moins d’un siècle il aura eu le temps de s’intéresser à l’analyse (l’essentiel de ses travaux), à l’algèbre, à la géométrie, à la théorie des nombres, aux probabilités sans parler de la physique et de la philosophie auxquelles il a consacré quelques articles.
Il meurt un 18 septembre à un âge qui n’est divisible ni par 3, ni par 5 ni par 7. L’année qui suit sa mort est une année bissextile dont la somme des chiffres est divisible par 5. A quel âge est mort Euler ? Quel jour de la semaine ?
*Sont bissextiles les années divisibles par 4 à l’exception des années multiples de 100 et non divisibles par 400. *Une anné normale a 365 jours et une année bissextile 366.
I think it says that Euler was born in Basle April 15, 1707. He died more than a half century later in Saint Petersburg, leaving a considerable mathematical legacy. In a life that lasted less than a century he had the chance to engage himself in analysis (his most important work), algebra, geometry, number theory, and probability, without even speaking of physics and philosophy, about each of which he wrote several articles. He died on September 18 at an age that was not divisible by 3, by 5, or by 7. The year after his death was a leap year, and the sum of the whose digits was divisible by 5.
At what age did Euler die? What day of the week?
Is this a good olympiad-style question? Seems strange to me, but that 3, 5, 7 business appeals.
JD2718 
First, I’ll clarify the problem slightly. In your translation, I wasn’t sure whether the year of his death was supposed to have a digit sum divisible by 5, or the year after his birth. Checking in the French, I see that it is the leap year. I would re-translate that sentence as:
The year after his death was a leap year, the sum of whose digits was divisible by 5.
Anyhow, onto the problem. Nothing cute here. I started checking leap years after 1757 for one with a digit sum divisible by 5. the first one I came up with was 1784. If Euler died in September of 1783, he’d have been 76 years old. 76 = 2^2*19 so it is not divisible by 3, 5, or 7 and meets the requirements of the problem.
So, he died at age 76 on September 18, 1783.
Knowing how to calculate the day of the week from a given date seems like a pretty esoteric skill. I looked it up on Wikipedia. They happen to have used September 18, 1783 as one of the examples (leading me to feel even more confident that my answer to the first part of the question was correct!) and it was apparently a Thursday.
And, no, I don’t really like it as an “Olympaid” question. Particularly the day of the week thing. The algorithm is nothing I would want to encourage students to memorize, and figuring it out by hand based on a known current era date seems quite messy and un-fun.
I got annoyed before I got as far as mathmom… So, no, I don’t think its a great Olympiad question.
That website had some really nice problems, but I think I agree with both of you – this isn’t one of them.
All the same, I was pleased I could read math in French. Last took a course in 1981….
The day of the week computation is pretty easy. Ordinary years have 5 weeks and one extra day, so a given date advances one day of the week per year. But leap years have two extra days, so a date in a leap year after February ) advances two days of the week. September 18 2007 is 224 years after September 18, 1783, so ignoring leap years, the day of the week should have advanced 224 days, but there have also been 54 leap years in the interval (year numbers divisible by 4 except 1900 and 1800, which are not divisible by 400) so the total advance is 278 days or weeks and 5 days. So September 18, 2007 (a Tuesday) is 5 week days ahead of that date in 1783, and Tuesday is 5 week days ahead of Thursday. Of course this only works if we are looking at dates on the Gregorian calendar, so going back to earlier dates requires getting onto the Julian calendar dating system.