got discuss before already.
http://chitchat.sgforums.com/?action=thread_display&thread_id=170563all use the intrinstic properties of the number 9.
u help me solve this lah:
www.ebaumsworld.com/magic/choosenumber.htmlPROOF:
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 <= 9With 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.
