Here is a scan of my A5 sheet workings.
Years are down the left margin
Next columns are the prime factors for each year (helps to find early/late dates)
Followed by the earliest possible date
then the latest possible date for each year.
Finally a tally of the number of days between the two Magic Dates
I “looped” the gaps between dates to help keep track of what I was doing.
I only did the counting of gaps that looked promising.
The other side of the sheet has the years 27-55 and 96-00
Wow! That’s much more robust and thorough than my approach!
I guessed that there would be a 3 or 4 calendar year difference, say with 1 or 2 prime years and an invalid co-prime year in-between. A co-prime is the product of 2 primes, e.g. 43 and 2. I was lucky in spotting the 1097 days example since I started close to my real date of birth! I didn’t cover the whole century.
Yes, it was that string of three years 57, 58 and 59 where I noticed the first two were co-primes and the third was a Prime. And especially 58, with 29 and it not being a leap year. I had the 1097 for quite a while, but it took a bit of methodical work to ensure there weren’t any other similar combinations.
Nice teaser.
Three coins in the fountain … well, not exactly !
A table has 4 coins placed on it with two coins facing heads-up and two coins facing heads-down.
You were previously blindfolded, and still are, so can’t see the coins.
How can you re-arrange the coins into two piles such that each pile has the same number of coins showing heads-up ?
I would add that the process isn’t limited to just 4 coins.
It should work with 40 coins (in other words, any multiple of 4 coins, I think) provided the starting point is that half the coins are heads-up and half the coins are heads-down.
I’ve only tried it up to 16 coins and it impresses the grandchildren 
You can do it with gloves on if they think you can feel the difference between heads/tails with your fingers - you don’t need to feel the difference.
Are you allowed to turn any coins over at any time?
Hi Steve, Yes.
You can turn as many coins over as you wish.
In that case, step1 is to split the coins into two pairs, then step2 is to take one of the pairs and turn both coins over.
Well done Steve, and nicely described !
As a party-piece with the grandchildren, I normally use 16 coins and wear a pair of gloves - they are convinced that I can feel heads and tails. Add in a few mumbo- jumbo words and, hey-presto …. they are amazed.
I like the idea of all the theatrics!
As an aside, I think it works for any even number of coins?
I think you are right Ravvie. Well, more accurately, you ARE right.
Never realised that until you pointed it out. Thank you.
All you have to do for this one, is find the numbers that fit the boxes.
Normal arithmetic, no tricks, just straight forward arithmetic. (or, if it helps, algebra !)
I make it that the solution involves non-integers.
that is what I found.
One of those that at first glance looks very easy. I tried to solve it in my head but eventually had to resort to equations 
Same story here Steve ….
I’ve filled up two pages of scrap paper with scribbles last night and now having respite care.
I know the feeling Mike !
I filled half a page with trial and error guesses, then opted for algebra. Even then, I had to work methodically.
But all came good in the end. Hope the respite care goes well
For the sake of completeness, my solutions were as follows. Top line 3.5 and 4.5. Second line 9.5 and 3.5.
