# Rep Digit Palindrome expressed as a Unique set of Pan Digital Relations

Allowed Operations:

Addition, Subtraction, Multiplication, Division, Exponentiation. Right and Left Bracket use permitted.

Express $$2222$$ in $$2$$ different ways using all the digits 1 to 9 only once in each expression.

$$2222$$ =

$$2222$$ =

• So you can't use zero?
– Duck
May 31, 2019 at 2:23
• No..only 1 to 9,,all need to be used without repeating
– Uvc
May 31, 2019 at 2:28
• And also, can you make, for example, 4 and 5, 45?
– Duck
May 31, 2019 at 2:29
• If you mean it as forty five..sure
– Uvc
May 31, 2019 at 2:31
• For example: one of the terms could be..(5/1) ^ 3
– Uvc
May 31, 2019 at 2:36

I'm going to try:

$$2222=1987+234-5+6$$

and

$$2222=1986+234-5+7$$

Just a little bit after @JonMark Perry

$$(49+52) * (8+3) * (7-6+1)$$

and technically different

$$(42+59) * (3+8) * (7-6+1)$$

• Good one.........
– Uvc
May 31, 2019 at 18:48
• Just some prime factorization...
– Duck
May 31, 2019 at 22:20

Solution 1:

To get a “big” number like $$2222$$, I decided to use powers.

$$2^{11}=2048$$ is close to $$2222$$ but is a bit too small.
I need to increase my solution by $$2222-2048=174$$.
I can express the power of $$11$$ as $$3+8$$ or $$4+7$$ or $$5+6$$.

Examining the $$3+8$$ case, the unused digits are: $$1,4,5,6,7,9$$.
We can easily make the number $$174$$ using the digits $$1,7,4$$.

To deal with the remaining digits $$5,6,9$$ we use two facts:
$$1^n=1$$ and multiplying by $$1$$ doesn’t change the value of an expression.

Finally, we get:

$$2222=2^{3+8}+174 \times (6-5)^9$$

Solution 2:

Considering powers of $$3$$, I got a nicer solution:

$$2222=3^7+1+2+4+5+6+8+9$$