# Representation of 3 with Three Equal Digits

Use operations of addition, subtraction, multiplication, division, square root, and the decimal point to make true the identities below.

In other words, represent $3$ with three instances of each digit. For example $1+1+1=3.$

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Copyright © 1996 - 2013 Alexander Bogomolny

### Solution

$1+1+1=3$

$2+2/2=3$

$3+3-3=3\cdot 3/3=3$

$4-4/4=3$

$.5\cdot 5+.5=3$

$.6\cdot 6-.6=3$

$7/.7-7=3$

$\sqrt{8+8/8}=3$

$\sqrt{9}+\sqrt{9}-\sqrt{9}=3$

It was observed by a 16 year old boy from Israel that there is an universal template that works for all non-zero decimal digits: $\displaystyle\sqrt{(X-.X)/.X}=3.$

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Copyright © 1996 - 2013 Alexander Bogomolny

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