   SEARCH HOME Math Central Quandaries & Queries  Question from Julie, a parent: Hi, please help me!!! I have a subtraction puzzle... 5 digits minus 4 digits = 33333 Only using numbers 1-9 once each. I can get 33323 but then thats it!!! Please how do I work it out? There must be a simple way?? Thankyou Julie,

There is no simple'' way, it is a puzzle to make you think about subtraction. My first thought is that the leftmost digit of the five digit number must be a 3 if there is no borrowing, or a 4 if there is borrowing. The simplest hypothesis is no borrowing: 3**** - **** = 33333.

The simplest solution would be digitwise, with no borrowing at all:
4 - 1 in one position, 5 - 2 in another position, something like 345** - 12** = 33333. But the remaining digits are 6, 7, 8, 9, and there is no way to arrange them in pairs with difference of three.

So, some borrowing has to occur, which means that the digits will be matched in pairs with a difference of
three (like 5 and 2) and sometimes a difference of 4 (like 5 and 9), when a borrowing occurs. Note that I view the differences cyclically, 9 could be matched to 2, because 12 - 9 = 3.

At this point, what helps is to make a diagram, linking pairs of digits that cannot be matched: 1 cannot be matched to 2, 3, 6, 9, and 2 cannot be matched to 1, 3, 4, 7 ... There we begin to see that we could match 2 and 5, 6 and 9, 7 and 3 and 1 and 8; the remaining number is 4, which would have to be the leftmost digit of the 5 digit number. It almost works, but not quite. with 41597 - 8263, we get 33334, but if we force a borrow from 7, as in 47159 - 3826, we don't borrow from 4 anymore, so we get 43333. Can you see how to fix it?

Claude     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.