Creative Puzzle Thinking [Nob Yoshigahara 1993, problem 1]:
Problem: Even Odd
Addition
An odd number plus an odd number is an even number.
An even number plus an odd number is an odd number.
An even number plus an even number is an even number.
Multiplication
An odd number times an odd number is an odd number.
An even number times an odd number is an even number.
!!An even number times an even number is an odd number!!
In what context makes this sense?
Property: no zero-like number
An number z is zero-like if z + z = z and z * z = z.
Given a set S with two operations + and *.
Each element of S is either even or odd.
The operations have the properties
odd + odd = even,
even * even = odd.
Then there is no zero-like element in S.
Proof: Assume there were a zero-like z.
Is z now even or odd according to the rules?
Case: let z be even
As even * even is odd we have z must be odd.
Case: let z be odd
As odd + odd is even we have z must be even.
Therefore z can not be even or odd.
So there is no algebraic solution to this puzzle.
Solution: Nob Yoshigahara
Count numbers of letters
"An odd number plus an odd number"... 26 (even) letters
"An even number plus an odd number"... 27 (odd) letters
"An even number plus an even number"... 28 (even) letters
"An odd number times an odd number"... 33 (odd) letters
"An odd number times an even number"... 34 (even) letters
But: "An even number times an even number"... 35 (odd) letters !!
And more, "even number" is even number and "odd number" is odd number.
http://ux01.so-net.ne.jp/~nobnet/CPT/
References:
- Nob Yoshigahara;
Creative Puzzle Thinking,
Atlanta (Benjamin May High School), 1993-01-11. (11 puzzle questions)
reprinted:
Elwyn Berlekamp and Tom Rodgers;
The Mathemagician and Pied Puzzler,
1999, p37-40 (11 puzzle questions, but not all are the same)
- Nob Yoshigahara;
Solutions of Creative Puzzle Thinking,
http://ux01.so-net.ne.jp/~nobnet/CPT/
Appendix:
An Even Odd Bewilderment [Email of Fred Lunnon to math-fun at 2000-02-11]
I can't cast any light on Nob's puzzle; but it reminds me of a hilariously
deep pit that, when my children were teenagers, a school mathematics teacher
of my acquaintance unsuspectingly dug for himself. The occasion was a gathering
of pupils' parents, to whom he had undertaken to explain --- in simple layman's
terms --- the "New Maths" which was in the process of being introduced into the
secondary syllabus. The topic he had selected was the arithmetic of the binary
field.
He had not unreasonably decided that overloading the meanings of "0" and "1"
would be liable to cause confusion with ordinary integers, and therefore
explained (in, I fancied, a somewhat condescending fashion) that he would
simply abbreviate "even" to "E" and "odd" to "O"; following which he brought
forth with a magisterial flourish his prepared OHP slide of the ensuing
addition and multiplication tables:
+ E O x E O
------ ------
E | E O E | E E
O | O E O | E O
At this point, his undergraduate training re-asserted itself in a spectacular
fashion: "e" of course, customarily denotes the unit element of a ring,
and "o" the zero. In that order. "Oh dear", exclaimed the prophet, as he
inspected the first entry, "that's not right: it should be "E+E=O, ha-ha."
Amusement turned rapidly to consternation as he inspected and amended several
further entries, and finally to perspiring and uncomprehending panic as the
magnitude of the disaster began to dawn on him. Gradually, an expectant silence
descended upon his numerous and tightly-packed audience.
Meanwhile at the rear of the hall, I eventually succeeded in controlling my
incipient hilarity sufficiently to interrupt and essay a clarification
of the difficulty: but of course in such circumstances it is well-nigh
impossible for a lecturer to switch from output to input mode well enough
to grasp what day of the week it is, let alone to absorb a linguistic conundrum
of this subtlety. On the (fortunately rare) occasions when we subsequently
encountered one another, he eyed me with the air of one suspecting a warlock
of having recently cast a particularly nasty spell; and I can't refrain from
the speculation that, as a result of this single traumatic incident, we may
have been responsible for sabotaging the entire campaign to reform mathematics
teaching throughout the United Kingdom.
Fred Lunnon
Questionable Mathematics.
"Question: Name one thing beneath Bill Clinton's dignity.
Answer: This is a trick question, like asking whether zero is odd or even.
It has no known answer."
Zero is even of course.
--
http://www.mathematik.uni-bielefeld.de/~sillke/
mailto:Torsten.Sillke@uni-bielefeld.de