got discuss before already.
http://chitchat.sgforums.com/?action=thread_display&thread_id=170563
all use the intrinstic properties of the number 9.
u help me solve this lah: www.ebaumsworld.com/magic/choosenumber.html
PROOF:
Mod sign (triple-dash hyphen) is indicated as '='.
Let N be the original number.
For some N is a member of the real number set,
N
= 10³·a + 10²·b + 10¹·c + 10·d
= 999a + 99b + 9c + (a + b + c + d)
= (a + b + c + d) mod 9
>>>
Since 999k, 99k or 9k are all multiples of 9
>>> This effectively means the remainder when N is divided
by 9 is equal to the sum of all its digits. And N can be any real
number, regardless of how many digits it has, so this trick need
not be limited to 3 or 4 digits but any number of
digits.
For some number M which has the same digits (a, b, c, d) as the
abovementioned N, in any order, the same remainer can be obtained
when its constituent digits are divided by 9, i.e.
M
= 10³·b + 10²·c + 10¹·d + 10·a
= 999b + 99c + 9d + (b + c + d + a)
= (b + c + d + a) mod 9
N - M = [ (a + b + c + d) - (b + c + d + a) ] mod 9 = 0 mod 9
(N - M)
leaves a remainder of 0 when divided by 9, i.e. (N - M) is a
multiple of 9.
Now, we know for any number which is a multiple of 9, the sum of
its constituent digits is also a multiple of 9. Deducing from this,
a missing digit can easily be derived by obtaining the sum of the
other constituent digits and subtracting from the next highest
multiple of 9.
Proof:
For some real number P which is a multiple of 9,
P
= 10³·a + 10²·b + 10¹·c + 10·d
= (a + b + c + d) mod 9
= 0 mod 9 (since P is a multiple of 9)
a + b + c + d = 0 mod 9
>>>
(a + b + c + d) is a multiple of 9.
a + b + c + d = 9k, where k is some real integer
Let a be the missing digit.
a = 9k - (b + c + d)
>>>
The missing digit can be obtained by subtracting the sum of the
other constituent digits from a multiple of 9 such that 1 <= a
<= 9
With regards to not choosing 0 as the missing digit, it is due to
an overlapping result, i.e. a remainder of 0 or 9 may be obtained
for certain values of P, yielding 2 different results. Hence, a
certain set of values have to be defined for the missing value. In
this case, it is either
1. 0 <= a <= 8, or
2. 1 <= a <= 9.
