Hi Ravvie,
An hour or so ago I made a post showing Pete two ways to solve the problem. I thought it might have spoiled the teaser for others so deleted it.
The first option was to create two equations. One for their current ages the other for their ages in six years time. Two simple simultaneous equations.
The other was to start with two ages 28 years apart eg 29 and 1. Just keep adding one year at a time … and I guess this is what you are recommending ?
Not really. I just went straight to the solution assuming we were at six years then subtracted six
Ah !
Well done.
I used the simultaneous equations. But when drafting my explanation for Pete, I included the alternative of starting with 29:1 and going up in pairs until the first number became three times as big as the second number. I wasn’t able to imagine 42:14 straight away.
A Beast of a Puzzle
I was updating my profile today when I noticed that I have now received a total of 666 likes, the last one being from Don. OK, it is not a major milestone. However, Iron Maiden (and others) have stated that 666 is the Number of the Beast, hence the following puzzle.
The following summation has 666 terms, with the final term having 666 digits:
6 + 66 + 666 + 6,666 + 66,666 + 666,666 +…
What is the total?
It looks like an awful lot of 740’s
Rick s 36 now, and Roy is 8
in six years time, Rick will be 42 and Roy will be 14.
Nice teaser Pete, Thank you.
You get the cigar and the bottle of bubbly. 
Yes, but not quite
I realised it wasn’t quite all 740’s. The last three digits do follow some kind of sequence, but I think you are going to have to reveal the solution, unless someone else posts fairly soon.
Quite a puzzle. Nice one Ravvie.
How about I give a few hints, at least as to how I solved it:
A lot of terms with very long numbers does indeed suggest that there will have to be a pattern for it to solvable. I usually experiment using Excel.
As Excel can only cope with 15 digits accuracy, it is best to limit to the first 15 terms to look for a pattern.
As Don has noted, 740 appears quite a bit, but the last 3 digits is always a bit less. The pattern repeats every three terms.
I hope that is enough for mathematicians and non-mathematicians to have a go.
It can also be solved without Excel, needing only minimal maths (arithmetic), but it does require some lateral thinking.
Yes, I had noticed that the last three digits are always a bit less, and they sort of repeat every three terms.
The last 3 digits in the first three terms are 006, 072 and 738,
The next three terms do develop a bit of a pattern, with 404, 070 and 736
and we can see bit more of a pattern with the last digit of each set of three terms reducing by “2” eg 072, 068, 066 and 064 …
However, as I said above, i don’t see myself solving this one, so unless someone else can now come up with an explanation, I think Ravvie, you are going to have to reveal the solution.
But it is an interesting teaser ! Well done !
Thank you Pete 
FWIW, I also noticed that the last digit runs in a 5 entry sequence 6, 2, 8, 4, 0
The penultimate digit has a three term sequence, 0, 7, 3
And the third from last digit has a three term sequence 7, 4, 0 but only once you have passed the first term.
Interesting !
Beastly Solutions
I like the 666 puzzle as it is simple to state, looks horrendously challenging, yet there are a variety of ways to solve it. Here are a few:
1. Experimental Maths
If using a spreadsheet, note that each term is 10 times the previous term plus 6. Summing and considering every third result:
3 terms = 738
6 terms = 740,736
9 terms = 740,740,734
12 terms = 740,740,740,732
15 terms = 740,740,740,740,730
3n terms = n-1 groups of 740 followed by 740 - 2n
Hence if 3n = 666, we predict that the sum will be 221 groups of 740 followed by 740 - 2x222 = 296
My souped-up version of Excel (xNumbers add-in) confirms this is correct.
2. Lateral Thinking
Unless anyone is interested I will skip how I spotted this, but re-write each term as follows:
6 = 20/3 - 2/3
66 = 200/3 - 2/3
666 = 2000/3 - 2/3 etc
Summing the first term is easy, for example after 3 terms it is 2,220/3 = 740 and after 6 terms it is 2,222,220/3 = 740,740 etc.
The “- 2/3” term explains the reduction we saw empirically in the first solution.
3. Pure Maths
There are a few solutions online, all “mathsy”. Here’s an outline example of one of them:
Use algebra to create a geometric progression.
Substitute in the formula for a geometric progression to simplify.
Use more algebra to create another geometric progression.
Substitute in the formula for a geometric progression to simplify.
Use more algebra to create a general formula.
Calculate the answer.
Even though I used geometric progressions throughout my career in finance, my reaction was “Yuck!”
However, when I gave the puzzle verbally to Mrs R, after a few moments of pondering she said “Hmm, I reckon I should manipulate it to form a geometric progression.” She has been a bit too busy to try it though. Just goes to show how differently mathematicians can think from each other.
Nice teaser Ravvie.
And nice explanations also.
I was walking home the other evening along with my grandson’s maths teacher. I said to the maths teacher “Do you have any children yourself ?”
“Yes, said the maths teacher, I have three sons”
I asked “What are their ages ?”
The maths teacher told me “The product of their ages is 36”
I replied “I can’t tell their ages from that”
The maths teacher then said “The sum of their ages is one more than the number of the house we are passing”
Again, I said “I can’t tell their ages from that”
The maths teacher added “The eldest has just started piano lessons”
And with that information, I immediately knew their ages.
Any ideas ?
Note: There are three mathematical answers. However, two of these three possible answers involve highly improbable family settings of three children. The third answer strikes me as being a slightly unusual family, but is the answer given when I first encountered this teaser.
2, 3 and 6 works?
Hi Mike,
Good start, 2, 3 and 6 fits the first condition.
However, from the range of possible age-combinations, the sum of that combination (11) is unique, as are some of the other possible age combinations.
When passing a specific house number, I would have been able to recognise any such unique age-combination. I couldn’t.
Hope this helps.
Hmm. I need to do a matrix and omit the twins….
That’s it !
The matrix is really just a list of the possible factors of 36 that form triplets eg 36; 1; 1
There are only about half a dozen sets of triple factors to consider.