Repeated digits: five 5s and six 6s

How many results can you make by repeating just one digit? Try to make 5 with exactly five 5s, or 6 with exactly six 6s. Reaching the target is only half the task: you must also use precisely the required number of digits. The repeated-digits laboratory has forty ready-made targets and a workspace for creating your own.

Agree on the rules first

You may use addition, subtraction, multiplication, division and parentheses. You may also join digits: 55 counts as two 5s, while 555 counts as three. In the laboratory you can turn this option off for a stricter challenge. Powers, roots, factorials and decimal points are not allowed: the point is to reason with the four arithmetic operations. Every written digit must be the chosen one, used exactly the required number of times.

Three useful tools: zero, one and parentheses

The difference 5−5 is zero and uses two digits; the quotient 5÷5 is one. These are useful building blocks because A+0=A and A×1=A. For example, 5+(5−5)+(5−5)=5 uses exactly five 5s. Likewise, 6+(6−6)×(6+6−6)=6 uses exactly six 6s. Parentheses make the order of operations explicit: without them the same line of symbols can give a different result.

A five-5 example, step by step

Suppose the target is 10. The expression 5+5+(5−5)×5 contains five 5s. First calculate 5−5=0, then 0×5=0, and finally 5+5+0=10. If the target were 1, you could use (5÷5)+(5−5)×5=1: division makes one, while the remaining part makes zero.

A six-6 example, step by step

Suppose the target is 12. Write 6+6+(6−6)×(6−6). Both differences are zero; their product is zero; 6+6=12 remains. There are exactly six 6s. To make 0, one possibility is (6−6)+(6−6)+(6−6)=0.

Joining digits changes the game

Writing 55 does not introduce a new symbol: it uses the digit 5 twice to form a two-digit number. This can open new paths, but you must keep counting digits: 55÷5=11 uses three 5s, not two. That is why the laboratory checks both the exact value and the digit count. Division is checked as a fraction, with no rounding errors.

From puzzle to strategy

1. Write down the target and count the available digits.
2. First make a value close to the target with one or two digits.
3. Use d−d or d÷d to spend remaining digits without changing the value, when possible.
4. If that fails, change the operation order or form a number such as 55 or 66.
5. Always check both the result and the digit count.

Try the laboratory

The interactive laboratory offers twenty targets for five 5s and twenty for six 6s. Choose a target, enter an expression and press “Check”. You can ask for a hint or reveal a verified solution. In the custom workspace, choose a digit from 1 to 9, four to six uses, and a target between −100 and 100. Not every invented target is necessarily reachable: discovering why a route fails is part of the game too.