Well spotted that it involves a geometric progression. The following works for any value of r bigger than 1 (r=5 in the teaser):
X/r + X/r^2 + X/r^3 + X/r^4 +…. = X/(1-r)
The above formula continues to infinity and without discarding any fractional remainders. This is why our total is slightly less than X/4.
The algorithm is equivalent to converting X from decimal to base 5:
100/5 = 20 R 0
20/5 = 4 R 0
4/5 = 0 R 4
This leads to 100 = 400(base 5)
The 20+4+0 is the answer we have already. The “slightly less” is just the sum of the remainders divided by 4. So we could use 25 - (0+0+4)/4 = 24 as an alternative.
Consider X = 624:
624/5 = 124 R 4
124/5 = 24 R 4
24/5 = 4 R 4
4/5 = 0 R 4
Hence 624 = 4444 (base 5) leading to “slightly less” = (4+4+4+4)/4 = 4.
This gives 624/4 - 4 = 156 - 4 = 152, which is the same as 124+24+4.
If we want a formula for the number of prime factors r in any X!:
Total = (X - d)/(r-1), where d is the sum of digits of X(base r)
My two grandsons, Stephen and Thomas, had been enjoying ice-creams. I asked them to look at me, and I said, “At least one of you has got ice-cream all over your face !”
They looked at each other somewhat surprised ! Each could see the other’s face but not their own.
I then said, “Kindly step forward if you have got ice-cream all over your face”. Neither of them moved!
Again, I said “Kindly step forward if you have got ice-cream all over your face”. What happened next, and why ?
You can assume that children, even naughty children, have the analytical skill of a professional logician, and in addition behave honestly all of the time !
Despite getting A’ level maths at aged 16 (50 years ago), I have to admit that this level of work is completely beyond my understanding and comprehension.
I envy both you and Don!
Just out of interest, what level of maths studying did you both get to?
They both stepped forward?
At the first iteration, each could see the other’s messy face, but of course were unsure about their own.
At the second iteration, each deduced that they themselves must have ice cream on their own face as well, realising they must both have been looking at the same thing.
I got a degree in Maths then further professional training, becoming an actuary.
The maths I tend to use in practice (for teasers etc) is typically based on that taught up to A-level. However, such “school maths” tends to be the building blocks, usually taught as individual techniques used in specific circumstances.
Since school, I learned how to apply the techniques, often combining them. This means I continue to develop my understanding, hence I tend to spot relationships.
This teaser was hard, given it required spotting both geometric progressions and base conversions. Hardly obvious, but it did fit in with my preferred “experimental mathematician” approach.
When I solve maths problems, I see the objective as “why is that so”, rather than just “what is the answer”. There’s a risk that it can be a bit “mathsy”, but I hope that’s not too off-putting.
“experimental mathematician”
That’s the perfect description of how I’ve observed your approach over the years, particularly on the harder teasers.
I know a few actuaries, and the quality maths background is a common factor (pardon the pun!).
I’m an accountant, and know my maths/logical thinking is better than most, but as I said before, I am envious of people who see very complicated things very clearly and/or can think outside the box.
A bit like “Only Connect” if you’ve ever watched it.
At A level I did Pure Maths, Applied Maths, Physics and Chemistry.
At University I did a BSc in Civil Engineering which incorporated a bit more maths during the first two years, which was useful in designing/analysing complex structures. The engineering elements of the course certainly encouraged creative thinking as well as rationalising and simplifying problems and their solutions.
I spent about five years (‘68 - 73) in The Gulf, mainly Kuwait, Trucial States (UAE now) and Oman. Thursday afternoons and Fridays included a bit of time on a beach doing the Sunday Times’ “Brain Teasers”.
More recently, i’ve been helping my eldest grandson revise for his up-coming GCSE Maths. Some of the more recent teasers have been based on his revision. I’m often surprised at how much i’ve forgotten and need to revise myself
Nevertheless, I’ve tried to recall a broad range of teasers to share on the Forum, and been delighted at the many teasers posted by others. Two that always spring to my mind are “The Ladder” posted by ‘BAM’ way back c.2001 and ‘water-level in a small lake’ posted about the same time. Neither require more than A Level Maths, but they certainly require careful thinking !! However, there are hundreds more and each one has been a delight for me and hopefully many others.
Don (and everyone else!), your efforts on this thread are much appreciated. I don’t post on here very often, but I really enjoy following and thinking about the teasers. Thank you!
Thank you Bobby, very thoughtful of you and really appreciated.
Let’s hope that between us all there are still a few teasers left in the cupboard to keep us amused, and a few of the old grey cells working !
When my 8 year old grandson asked me that question, he made it clear that at his school they don’t say, for example, “one hundred and one”. They simply say “ one hundred one”. Ie they don’t include the ‘and’ word!
After a short discussion, I conceded, and …
Counted again, excluding the ‘and’ word
