Problem:
For how many pairs of consecutive integers in is no carrying required when the two integers are added?
Solution:
Let have decimal representation . If one of , or is , or , then there will be carrying when and are added. If and , or if and either or , there will also be carrying when and are added.
If is not one of the integers described above, then has one of the forms
where . For such , no carrying will be needed when and are added. There are such values of .
The problems on this page are the property of the MAA's American Mathematics Competitions