What's 3 + 7? What's 4 + 6?

Additional material

[ Editor: a calculus version of the grey elephant from Denmark trick,
from a 1997 thread "math puzzle?".  Wayne Hacker first posted it with
an error; his corrected version: ]

(1) Pick a and b with a<=b (real numbers)

(2) m = (b+a)/2

(3) u = sqrt(3)(b-a)/6

(4) c = m - u

(5) d = m + u

(6) Pick f(x), cubic with real coefficients

(7) e = f(c) + f(d)

(8) x = 1 + [ {area under f from a to b}/(u*e) ]^2

(9) convert x to a corresponding letter by choosing the nearest integer
(mod 26) to x (1=a, 2=b etc).

(10) think of a country starting with your letter, animal with next
letter and a color in which the animal is found.

-- Wayne Hacker <bhacker@cise.ufl.edu>

[ Editor: the first version had u=(b-a)/6, c=m-sqrt(u), d=m+sqrt(u) and
x = 1 + {area}/(u*e) without the square, and ended "there are no grey
elephants in Denmark".  Two replies to it: ]

First time I chose a = 2, b = 8, and f(x) = x^3.
Therefore m = 5, u = 1, c = 4, d = 6, and e = 280.
The area is (b^4-a^4)/4, and thus it is 1020.
Finally int(1+1020/(280*1)) = 4, and I ended up with a grey emu from
Dahomey.

Second time I chose a = 2, b = 20, and f(x) = x^3.
Therefore m = 11, u = 3, c = 9.26794919243, d = 12.7320508076, and e = 2860.
The area is (b^4-a^4)/4, and thus it is 39996.
Finally int(1+39996/(2860*3)) = 5, and I ended up with a pink flamingo from
Ethiopia.

Third time I chose a = 2, b = 200, and f(x) = x^3.
Therefore m = 101, u = 33, c = 95.2554373535, d = 106.744562647, and
e = 2080600.
The area is (b^4-a^4)/4, and thus it is 399999996.
Finally int(1+399999996/(2080600*33)) = 6, and I ended up with a white goat
from Finland.

-- Ilan L Mayer

A slightly more high-tech version of an old trick (personally, I prefer
the version where you choose an animal beginning with the second letter
of the country you chose).  But unless I am somewhat mistaken those values
should be:

c = m - sqrt(3)*u
d = m + sqrt(3)*u

For example, without this correction choosing b = 2, a = -1 and x^2 as the
original polynomial could lead to a silver fish from Egypt, or a fly, or
various other animals.

-- Geoff Bailey <ftww@cs.su.oz.au>

[ Editor: why the corrected version always gives 4 (D): c and d are the
two-point Gauss-Legendre nodes, which integrate any cubic exactly, so
the area is (b-a)/2 * e.  With u = sqrt(3)(b-a)/6 the ratio
area/(u*e) is sqrt(3), its square is 3, and x = 4.  Bailey's correction
(with u = (b-a)/6) gives area/(u*e) = 3 and x = 4 without squaring. ]

Choose a two dimensional shape, for example a square or a circle.
    They usually choose a triangle.

-- Shlomo Hershkop <sh272@pipeline.com>

I read this in a magic book, aeons ago... I forgot which one.  It
doesn't always work, but when it does it's a jawdropper.  The exact
phrasing is important.  Works better than half of the time.  In a group,
you can tell people that you're going to project the number into your
mind; then the trick "works" if most of the people pick the number you
are expecting, even if it's not all of them.

"Think of a number less than fifty both of whose digits are odd, but are
not the same.  For example, you cannot choose eleven, because although
both of its digits are odd, they are the same."

Got one?  Good... scroll down a bit...













The number you picked was:  THIRTY-SEVEN!

Well, if it didn't work, you're not impressed.  (In which case, well,
wait... let me focus the mental rays... uh, the number you picked was
THIRTY-FIVE!)  If you are impressed, even momentarily, you'll soon
notice that actually you have only given them eight numbers to choose
from: 13, 15, 17, 19, 31, 35, 37, and 39.  Telling people that eleven is
illegal tends to direct their attention away from ALL of the teens, and
within the thirties... well, 37 is just the commonest choice by far,
with 35 running second.

-- Daniel P. B. Smith <dpbsmith@world.std.com>

[ Editor: a trick posted in 1996 by Saighiridhar Vijayagopalan; in the
original each line was followed by many blank lines. ]

> FOLLOW these instructions one at a time and as QUICKLY as you can!
>
> Then follow the instructions afterwards, ok?  Answer as QUICKLY as
> possible but don't advance until you've done each of them!!!
>
>   What is:
>
>   2+2?
>
>   4+4?
>
>   8+8?
>
>   16+16?
>
>   Quick! Pick a number between 12 and 5.  Got it?  Now page down...
>
>   The number you picked was 7 right?
>
>   Freaky?

I got 8 (but I'm a left-brained lefty for whom, in the words of William
Goldman, the shortest distance between two points is a parabola

I like Matthew Priestly's explanation.  I'd be interested in the
results if the question asked for a number between:

        (a)  "15 and 2"

    - or -

        (b)  "5 and 12"

-- Gary Scott Simon <garscosi@nyc.pipeline.com>, February 1996

[ Editor: Matthew Priestly's explanation is not in the collected
material. ]