I can multiply any three-digit number by 1667 in my head, usually in less than 10 seconds.

Also, to make it even more impressive, I come up with the digits of the result in proper order (not in reverse!)

How can I do this?

i posted this solution several days ago...maybe it is not clear, so here i go again.

(1)Let the given number x between 100 and 1000 be represented by the 3 digits abc (which means x = 100a + 10b + c).
(2)Mentally calculate the number 2x. Since x ranges from 101 to 999, 2x ranges from 202 to 1998 and hence has a minimum of 3 and a maximum of 4 digits. let the number 2x be represented by the 4 digits ABCD where the leading digit A = 0 if 2x is a 3-digit number. (this means 2x = 1000A + 100B + 10C + D where A = 0 if 2x has only 3 digits.)
(3)Juxtapose x with 2x on its right to come up with the 7-digit number
<x><2x> = <abc><ABCD> = abcABCD.
(4)Mentally divide this number by 6 and you get the answer from left to right. Multiplying the given 3-digit number mentally by 2, juxtaposing the given number on its right with twice the number (after adding a leading 0 if twice the number has only 3 digits), and dividing the resultant 7-digit number by 6 mentally can all be done under 10 seconds. For instance,
(a)1667*101 = 101<0202>/6 = 1010202/6 = 168367;
(b)1667*674 = 674<1348>/6 = 6741348/6 = 1123558; and
(c)1667*999 = 999<1998>/6 = 9991998/6 = 1665333.

Here is the basis for this method.
1667 = 10002/6.
So, 1667*x = 1/6*(10002x) = 1/6*(10000x+2x)
Since 202 ≤ 2x ≤ 1998, 2x has a minimum of 3 and a maximum of 4 digits, represented by the digits ABCD. 10000x has the number x followed by 4 zeroes at the end. so, it is a piece of cake to add 10000x to 2x to get the number 10002x. In terms of digits,
10000x = abc0000 (since x has the digits abc),
2x = 000ABCD, and
10002x = abcABCD
Hence abcABCD divided by 6 is equal to 10002x/6 or 1667x, which is the answer we are looking for.

Posted by mohan on 2003-10-13 04:45:20