A good book for short problems What is a good book for problems which can be done without much mathematical background? I don't mean IMO-level, since those questions generally require a fairly big amount of mathematical knowledge, but more like questions along the line of the toughest questions on a kangaroo test, some examples:
We call a natural number a funny number if it is the smallest of all numbers that have the same sum of their digits. 
For example: 29 is funny, because it is the smallest number of which the sum of its digits is 11.
We add up all funny three-digit numbers.
What is the outcome?
Mister Small, Mister Medium and Mister Large are going for a walk. 
Mister Large says: “Funny, how our names are all about lengths, but each of us has 
a wrong name.” The smallest of the three answers: “Yes, that is true”.
What are the names of the  gentlemen from smallest to largest?
If you multiply 1, 2, 3, …..up to n (inclusive), the result is $2^{15} \times 3^6 \times 5^3 \times 7^2 \times 11 \times 13$.
Which number is n?
So fun questions like these which aren't too long and can be purely done with a decent mathematical intuition.
 A: You might want to peruse this list and see if anything catches your eye a,d it hopefully gives you some ideas on other things to look for.


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*Entertaining Math Puzzles, Martin Gardner

*Mathematics, Magic and Mystery, Martin Gardner

*The Colossal Book of Mathematics, Martin Gardner

*The Last Recreations, Martin Gardner

*More Perplexing Puzzles and Tantalizing Teasers, MARTIN AUTOR GARDNER

*Classic Brainteasers, MARTIN AUTOR GARDNER

*The Moscow Puzzles, Boris A. Kordemsky

*Logic Puzs to Bend Yr Brain, Kurt Smith

*More Mathematical Puzzles of Sam Loyd

*Challenging Logic Puzzles, Barry R. Clarke

*Second Scientific American Book of Mathematical Puzzles and Diversions

*Hard-To-Solve Math Puzzles, Derrick Niederman

*A Passion for Mathematics, Clifford A. Pickover

*Problem Solving Through Recreational Mathematics, Bonnie Averbach and Orin Chein

*The Ultimate Book of Puzzles, Mathematical Diversions, and Brainteasers, Erwin Brecher

*25 Real Life Math Investigations That Will Astound Teachers and Students by Edward Zaccaro        

*50 Mathematical Puzzles and Problems Green Collection by Gilles Cohen (Editor)        

*50 Mathematical Puzzles and Problems Orange Collection by Gilles Cohen (Editor)   

*50 Mathematical Puzzles and Problems Red Collection by Gilles Cohen (Editor)  

*A Moscow Math Circle: Week-by-week Problem Sets by Sergey Dorichenko

*Chinese Brain Twisters by Baifang

*Challenging Critical Thinking Puzzles by Michael A. DiSpezio

*Great Critical Thinking Puzzles, by Michael A. DiSpezio

*Problem Solving Through Recreational Mathematics by Bonnie Averbach, Orin Chei
You might also like to explore Enjoying Math - Learning Problem Solving with Fun Math Puzzles
Regards
