# Equality and Fraternity

1. Consider the "equation" $6145 - 1 = 6143$. Can you move two digits so as to create a valid equality?
2. The equation $-127=-127$ is, of course, true. Suppose you were told again to move two digits as to leave a valid equation. Please note that you have to move two digits on the same side of this equation.

Note:

Moving means that you can change the position of any two digits independently, so that they still are a number/constant!

Hint:

The moving doesn't require swapping of digits in both.

It is a part of mathematical puzzles asked in the puzzling contest at my college(coordinated by me).

• Can you clarify what you mean by "move two digits"? Exchange their positions? Move them to any position? – frodoskywalker Feb 22 '16 at 18:36
• Ok, I am adding a hint. – ABcDexter Feb 22 '16 at 18:37
• That still doesn't quite answer the question - what exactly is permitted by "moving" digits? – DylanSp Feb 22 '16 at 18:44
• Anything, which keeps the mathematical meaning intact. – ABcDexter Feb 22 '16 at 18:57

First one:

$614-1^{54}=613$ or $615-1^{43}=614$

Second one:

$1-2^7=-127$

Lateral thinking says I can move these two digits in this way:

By using the $1$ as a line instead of a number,
1. $614 − 5 \ne 6143$
2. $-127 \ne 72$

• Not sure why the downvote. The "lateral-thinking" tag is applied, and I followed the instructions of moving digits. – Ian MacDonald Feb 23 '16 at 15:09
• I upvoted your answer, as it used to be a valid escape route to just make that tricky change into inequality... – ABcDexter Feb 24 '16 at 19:22

The second one is fairly easy (makes me think I might not quite understand where you can "move" a digit):

$2-17 = 2-17$ OR $1-27 = 1-27$ OR $-217 = -217$ or pretty much anything else like that.

Here's a solution to the first one that takes advantage of the vagueness of the instructions. I'm not sure if it's "legal" though.

Remove the $1$ from the $6145$ and put it sideways, turning it into a subtraction sign. Then move the $1$ from $6143$ over to the other side. You'll end up getting $645-1-1=643$.

• Good try, but please look at the updated question :-) – ABcDexter Feb 22 '16 at 18:53