Adding Numbers Cleverly
A list of numbers does not have to be added in the order it is written. Scan for pairs that make a round number and the arithmetic almost does itself.
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A list of numbers does not have to be added in the order it is written. Addition can be rearranged freely, so the first move should always be to scan for pairs that make a round number — then the arithmetic almost does itself.
Concept
Look for pairs first
Because addition can be done in any order, terms may be regrouped to suit you. Take :
Jumping to the first produces a
, and adding to a round number needs no thought. The targets worth hunting for are
,
and
.
Example
Pairs that make 100
Find .
Three hundreds: 
Look at the units digits to find these pairs quickly. Digits that add to — the
with the
, the
with the
— are the giveaway.
Example
Pairs that make 1000
Find .
Three thousands: 
The same trick, scaled up. Adding these six numbers left to right would take real effort; spotting the pairs turns it into counting to three.
Example
Decimals, with one left over
Find .
Pair from the outside inward, each pair making 
The middle value 
has no partner
With an odd count of terms one always sits unpaired in the middle. Add it on at the end rather than trying to force it into a pair.
Example
Squares first, then pairs
Find .
Work out the squares: 
Now hunt for convenient pairs among those results
Left over: 
and 
Here the pairs are not symmetric — you pair whichever results happen to combine neatly. The aim is round numbers, not a tidy pattern.
Summary
- Addition can be reordered freely, so never add blindly from left to right.
- Scan the list for pairs totalling
,
.
- Units digits that add to
- With an odd count of terms, one value is left unpaired — add it at the end.
- Evaluate powers or brackets first, then look for pairs among the results.
- Pairs need not be symmetric; any grouping that produces round numbers is fair.