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Brian has some boxes of paper clips . Some boxes hold 10 clips and some boxes hold 100. He has some paper clips left over. He has three more boxes with 100 paper clips than he has boxes with 10 paper clips. He has two fewer paper clips left over than he has numbers of boxes with 100 paper clips. What number of paper clips could he have.

• Well, the fewest number of boxes that qualify as “some boxes” is 1. And, if Brian has 3 more boxes with 100 than boxes with 10, the fewest boxes are those with 10. So, the fewest boxes of 10 can be 1. If Brian has 1 box of 10, then he has 4 boxes with 100. With 4 boxes of 100, he also has “two fewer” left over clips or 2. From all of this, the fewest number of total clips Brian can have is:

1 box of 10 = 10

+plus+

4 boxes of 100 = 400

+plus+

left over = 2

MINIMUM TOTAL = 412

After this minimum number, what is the minimum number of clips Brian can add and still satisfy all the conditions? Brian cannot add any more left over clips until adds at least one more box of 10. If he adds one more box of 10, he must also add 1 more box of 100. If he adds 1 more box of one hundred, he must also add 1 more left over clip. So, the minimum number of clips he can add after the 412 minimum is:

1 box of 10 = 10

+plus+

1 box of 100 = 100

+plus+

left over = 1

Brian must start with the MINIMUM TOTAL. But, he can add anywhere from 0 to X of the MINIMUM ADDITION amounts as long as “X” is any positive whole number (or zero):

POSSIBLE NUMBER OF CLIPS = 412 + 111X (where x is any positive whole number or 0)

+++++++++++++++++++++++++

Ooooooo, I just thought of something else. If Brian adds 8 more boxes of 10, he also adds 8 more boxes of 100. But, he also adds 8 more left over to the 2 from the MINIMUM AMOUNT. This would make the number of left overs equal 10 which he would add 1 more to the boxes of 10 and screw up everything. If he adds 9 more boxes of 10, he has a similar problem. In fact, none of the relationships given will hold after Brian adds more than 7 boxes of 10 to the first one. So, there are only 8 possible total number of clips Brian can have:

412, 523, 634, 745, 856, 967, 1078, 1189

Cool riddle!

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