Dude Numbers - Interesting Numbers Developed Following Henry Dudeney's Puzzle Number 114

Category:  Self Indulgence 

Henry Dudeney's fantastic book Amusements in Mathematics (available for free download) was published in 1917.  The book is a brilliant collection of over 400 brainteasers - fun for nerds!  Every puzzle presents a challenge; answers are provided - solutions are generally not discussed.  In my folder, I still have a bank of puzzles I cannot solve!

Puzzle 114 describes Curious Numbers; in this feature I wish to discuss this puzzle and generalise the idea to extend the challenge and curiosity.  

But first - what are curious numbers - I looked at Google Search and they directed me to the Penguin Dictionary of Curious & Interesting Numbers.  It seems there is a plethora of Curious Numbers - so we need to find a different name  - to acknowledge Henry Dudeney, I think Dude numbers is a good choice (I didn't like the other option of Dud numbers!).

Puzzle 114 is described in slide 1:

I really like the Dude numbers in slides 5 and  6 - both a and b come out to be the same numbers - amazing!  

Actually there is a pattern emerging from the study of these cases:

In slide 6, for the cube and square combination of the same digit p, one has  

                     m = p +1 and k = p  

N is then equal to p^3 - k or  p^3 - p = p(p+1)(p-1). 

Divide by m = p+1 and add p to get p^2 !!!

This algorithm may be used to build Dude numbers for all natural numbers.

I hope you have enjoyed reading this blog.  The possibilities of finding many more patterns are obvious  - let me know if you find any.

I am impressed by the genius of Henry Dudeney who had collected a fantastic range of puzzles.