# Something is wrong with the equation

There is something wrong with the equation below: By just exchanging two squares at a time, find the equality with the least amount of exchange.

For example if this question is asked for below equation, you may find the solution with one time exchange: the answer will be exchanging "+" and "9" which makes the equation correct as below:

$29+5=34$

FYI: Even though it is not a hint, all numbers are used only once. ($0,1,2,3,4,5,6,7,8,9$)

Hint: you only swap each square once for optimal clear solution.

• Do the squares we swap have to be adjacent like in the example? – Jaap Scherphuis Sep 5 '18 at 13:56
• @JaapScherphuis no, you can swap any square with any other square you want – Oray Sep 5 '18 at 14:05
• A few questions – (1) can I place the brackets beside a number (like $5(6)$) to do multiplication, (2) are we doing left-to-right or order of operations? – Hugh Sep 5 '18 at 14:23
• @Hugh left to right and you may yes – Oray Sep 5 '18 at 14:36

@oray's edit: Here is my intented answer since no-one found it out:

$3140\times(6+9)=2-75\times8$ (Initial layout)
$3140\times(6+9)-2=75\times8$ ("$=$" and "$-$")
$3\times40\times(6+9)-2=7518$ ("$\times$" and "$1$")
$3\times45\times(6+9)-2=7018$ ("$0$" and "$5$")
$3\times45\times(6+9)-7=2018$ ("$2$" and "$7$")

which is

the year we are in.

I found another solution with

4 exchanges

which is:

$3140\times(6+9)=2-75\times8$ (Initial layout)
$3+40\times(619)=2-75\times8$ (Swapped $1$ and $+$)
$3+40=(619)\times2-75\times8$ (Swapped $=$ and $\times$)
$3+40=(629)\times1-75\times8$ (Swapped $2$ and $1$)
$5+40=(629)\times1-73\times8$ (Swapped $5$ and $3$)
To arrive at $45=45$

————————————————————————————————————————————

Original post: I'm not sure if this is optimal, but I can do it in

4 exchanges

like this:

$3140\times(6+9)=2-75\times8$ (Initial layout)
$314\times0(6+9)=2-75\times8$ (Swapped 0 and $\times$)
$314\times0(6+9)=27-5\times8$ (Swapped 7 and $-$)
$314\times0(6+8)=27-5\times9$ (Swapped 8 and 9)
$514\times0(6+8)=27-3\times9$ (Swapped 3 and 5)
To arrive at $0=0$.

• this is good one, but not my intended answer and 4 is actually optimal though actual answer does not require any trick. – Oray Sep 5 '18 at 21:15
• @Oray what do you mean by "no trick"? this answer fits all the rules, does it not? – Quintec Sep 6 '18 at 2:02
• @Quintec originally i was not planning to accept 0( concept then i thought it would be okay :) – Oray Sep 6 '18 at 5:21
• According to a comment on the OP, evaluation should be left to right, which would give $27-3\times9=216$. FWIW, I'd prefer usual operator precedence in any arithmetic puzzle. – nickgard Sep 6 '18 at 8:06
• Hopefully the new solution I found was your intended answer @Oray :) – lovemathboy Sep 6 '18 at 8:54

I don't think it's optimal but I can do it in

$6$ exchanges.

Solution

Label the boxes, from left to right, by $A,B,C,\ldots,Q$. Then the exchanges are:

(i) $J \leftrightarrow M$
$3140\times(6+9 -=2)75\times8$

(ii) $L \leftrightarrow M$
$3140\times(6+9 -=)275\times8$

(iii) $K \leftrightarrow M$
$3140\times(6+9 -2)=75\times8$

(iv) $I \leftrightarrow K$
$3140\times(6+2 -9)=75\times8$

(v) $A \leftrightarrow I$
$2140\times(6+3 -9)=75\times8$

(vi) $D \leftrightarrow Q$
$2148\times(6+3 -9)=75\times0$

Well, to start it off, I can do

9 exchanges(thanks to @Jaap Scherphuis for reading the rules)

to end with

$31*0(69)+4=8*7-52$

the details of which I have not quite figured out how to display in the answer.

• Since non-adjacent exchanges are allowed, the above answer can actually be done in 9. It is the permutation (CJHIGFE)(LQ)(MOP) where the squares are labelled A to Q, which needs no more than 6+1+2=9 exchanges. – Jaap Scherphuis Sep 5 '18 at 14:51
• @JaapScherphuis thanks, my bad, didn't read the rules carefully – Quintec Sep 5 '18 at 17:34