What is my friend's PIN?

Question

Just a fun Puzzle:

My (fictive) friend told me his PIN: "My PIN is the highest 4 digit number that has the highest sum of unique differences."

He explained: "my PIN consists of 4 numbers A B C D.

• Calculate the absolute differences between A and B, A and C, A and D, B and C, B and D, C and D (so all differences)
• Remove duplicates (so each difference is included only once)
• Add the remaining differences together.

My PIN is the 4 digit number where this count is the highest and the highest of those numbers. "

What is my (fictive) friend's PIN?

Example:

The number 1469 has a count of:

1 ) calculate the differences: $ |1-4| = 3, |1-6| = 5, |1-9| = 8, |4-6| = 2, |4-9| = 5, |6-9| = 3 $

2) remove the duplicates: $ 3, 5, 8, 2 $

3) add them together: $ 18$

So if there is no 4 digit number with a count higher than 18 and 1469 is the highest number of these numbers then 1469 is my friend's pin. (but as I know both conditions are not met)

What is my friend's PIN?

In your answer please describe how you found his PIN (a clever way, or just brute force?)

Answer 1

9820

Reason:

4 numbers, they should have the biggest possible difference of it's members. So take 2 lowest and 2 highest possible numbers: 0, 1, 8 and 9.
But $|8-9|=1$, so let's take another number so that we can use all 6 differences. Next possibility would then be: 0, 1, 7 and 9(1) or 0, 2, 8 and 9(2)
Calculate the difference(1):
$|0-1| = 1, |0-7| = 7, |0-9| = 9, |1-7| = 6, |1-9| = 8, |7-9| = 2$
Numbers are: 1, 7, 9, 6, 8, 2. All 6 are different, so add them up and we get:
$1+7+9+6+8+2=33$
Calculate the difference(2):
$|0-2| = 2, |0-8| = 8, |0-9| = 9, |2-8| = 6, |2-9| = 7, |8-9| = 1$
Numbers are: 2, 8, 9, 6, 7, 1. We can see that differences are same as in 1, but just in different order. So if we add them up, we will still get $33$.
Now we should see which of those 4 is bigger: 0179 or 0289. To make it bigger we should re-order it in descending order, so we get:
1: 9710
2: 9820
and because $9820>9710$, we can say that his PIN code is indeed $9820$

Answer 2

(I assume that each of the four numbers $A,B,C,D$ is a digit in the range $0,\ldots,9$.)

Since the six differences do not depend on the ordering of $A,B,C,D$, let us assume without loss of generality that $A\le B\le C\le D$, so that the differences are

(1) $B-A$, $C-B$, $D-C$
(2) $C-A$, $D-B$, $D-A$.

The total contribution of the three values in (1) is at most their sum $(B-A)+(C-B)+(D-C)=D-A\le 9-0=9$. The total contribution of the three values in (2) is at most $9+8+7$ (in case they are all different; in case two are the same, the bound is even smaller). From these two upper bounds we conclude:

The count is at most $9+9+8+7=33$.

In order to reach this count, all inequalities in the above discussion must in fact be equalities. This implies $D-A=9$ and hence $A=0$ and $D=9$. Furthermore, the only ways of reaching differences $7$ and $8$ are

• $A=0$, $B=1$, $C=7$, $D=9$
• $A=0$, $B=2$, $C=8$, $D=9$

Together with these two possibilities, all their permutations are also feasible. Hence the highest possible corresponding number (and the pin number of your fictional friend) is the number 9820.

Answer 3

Just a brute-force answer; The answer is:

9820

This answer was generated by the following program:

$bigval = 0;
$biggest = array();
for($i=1000;$i<10000;$i++) {
    	$pin = (string)$i;
    	$diffs = array();
    	$diffs[] = abs($pin[0] - $pin[1]);
    	$diffs[] = abs($pin[0] - $pin[2]);
    	$diffs[] = abs($pin[0] - $pin[3]);
    	$diffs[] = abs($pin[1] - $pin[2]);
    	$diffs[] = abs($pin[1] - $pin[3]);
    	$diffs[] = abs($pin[2] - $pin[3]);
    	if(array_sum(array_unique($diffs)) > $bigval) {
    		$bigval = array_sum(array_unique($diffs));
    		$biggest = array($pin);
    	}
    	else if(array_sum(array_unique($diffs)) == $bigval) {
    		$biggest[] = $pin;	
    	}
    }
print_r($biggest);

Which gave the following output pins, with all the pins having a difference of 33:

1079, 1097, 1709, 1790, 1907, 1970, 2089, 2098, 2809, 2890, 2908, 2980, 7019, 7091, 7109, 7190, 7901, 7910, 8029, 8092, 8209, 8290, 8902, 8920, 9017, 9028, 9071, 9082, 9107, 9170, 9208, 9280, 9701, 9710, 9802, 9820

Of course the answer provided is the highest of these.

Answer 4

The answer is already found by Tryth and Novarg. So for variety sake adding my approach. 0, 9 and 8 are definite entrant due to range. Then I randomly chose 7 as remaining number and formed square:

   0  7  8  9
9  9  2  1
8  8  1
7  7
0

Giving 27.

   0  7  8  9
9  9  2  1
8  8  1
7  7
0

Next

   0  6  8  9
9  9  3  1
8  8  2
6  6
0

Giving 29 ... better.

Next

   0  5  8  9
9  9  4  1
8  8  3
5  5
0

Giving 30 better

Next

   0  4  8  9
9  9  5  1
8  8 [4]
4  4
0

Gone bad..

Next

   0  3  8  9
9  9  6  1
8  8  5
3  3
0

Giving 32 better

   0  2  8  9
9  9  7  1
8  8  6
2  2
0

Giving 33 and including 9 8 7 6 2 1

Going further down is going to repeat digits and getting 9 8 7 6 5 1 is not possible, so this one is the answer with largest number 9820

========================================================

Update (a more mathematical approach):

If in the descending order the digits are A>B>C>D

Then the differences are:
A-B,A-C,A-D;B-C,B-D;C-D
Adding all of them gives us:  
Total Sum= 3*A+B-C-3*D
Hence for the greatest sum, A & B should be as large as possible
and C & D should be as small as possible.
Which gives first winner combination 9810 but it fails as 9-8 = 1-0
So next one is 9820 which gives no overlap.