BrainDen.com - Brain Teasers
• 0

## Question

Take a pair of zeros and place them together.

00

Note there are zero digits between the pair of zeros.

Now place a pair of twos around them.

2002

Note there are two digits between the pair of twos, and still zero digits between the pair of zeros.

It's straightforward to construct strings with this property using [only] the remaining even digits.

420024

64200246

8642002468

Let's include the odd digits,  and eliminate the zeros.

For how many of these groups of digits, using the digits in pairs, can you construct strings with this property?

First solvers:

1

1,2

1,2,3 - 3 1 2 1 3 2 - CaptainEd

1,2,3,4 - 2 3 4 2 1 3 1 4 - KitnKat

1,2,3,4,5

1,2,3,4,5,6

1,2,3,4,5,6,7 - 61517346532472 - CaptainEd

1,2,3,4,5,6,7,8 - 8642572468531713 - hookemhorns

1,2,3,4,5,6,7,8,9

These by EventHorizon

11: 1,2,1,11,2,3,9,10,4,3,8,5,7,4,6,11,9,5,10,8,7,6

12: 1,2,1,3,2,12,10,3,11,4,5,9,6,8,4,7,5,10,12,6,11,9,8,7

15: 1,2,1,3,2,4,14,3,15,13,4,5,12,6,7,10,11,5,8,9,6,14,7,13,15,12,10,8,11,9

16: 1,2,1,3,2,4,16,3,13,5,4,15,12,14,6,5,7,8,11,9,10,6,13,16,7,12,8,15,14,9,11,10

1

1,2

1,2,3 - clarify: for this case, arrange 1 1 2 2 3 3 to have one digit between the 1's, two digits between the 2's and three digits between the 3's

1,2,3,4

1,2,3,4,5

1,2,3,4,5,6

1,2,3,4,5,6,7

1,2,3,4,5,6,7,8

1,2,3,4,5,6,7,8,9

Have fun.

Edited by bonanova
Eliminate the zeros from the sets of digits

## Recommended Posts

• 0
Take a pair of zeros and place them together.

00

Note there are zero digits between the pair of zeros.

Now place a pair of twos around them.

2002

Note there are two digits between the pair of twos, and still zero digits between the pair of zeros.

It's straightforward to construct strings with this property using [only] the remaining even digits.

420024

64200246

8642002468

Let's include the odd digits.

For how many of these groups of digits, using the digits in pairs, can you construct strings with this property?

0 - solved above

0,1

0,1,2

0,1,2,3

0,1,2,3,4

0,1,2,3,4,5

0,1,2,3,4,5,6

0,1,2,3,4,5,6,7

0,1,2,3,4,5,6,7,8

0,1,2,3,4,5,6,7,8,9

Have fun.

0

Then you put a pair of 1's around it and you have

101

Now there is one digit between the pair of 1s and zero digits between the single 0.

Or did you want to use two zeros?

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• 0

0

Then you put a pair of 1's around it and you have

101

Now there is one digit between the pair of 1s and zero digits between the single 0.

Or did you want to use two zeros?

Use the digits in pairs.

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• 0
Use the digits in pairs.

Ok, I'm dumb

1001 - TWO digits between the ones

210012 - TWO digits between the ones, FOUR digits between the twos.

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Ok, I'm dumb
1001 - TWO digits between the ones

210012 - TWO digits between the ones, FOUR digits between the twos.

First string fails because there is not ONE digit between the pair of ONEs.

Second string fails because there are not TWO digits between the pair of TWOs.

I am going to amend the OP to eliminate the use of zeros.

Thus, for the case of

1, 2, 3

we seek a string of six digits: 1 1 2 2 3 such that there are

one digit between the pair of 1s

two digits between the pair of 2s

three digits between the pair of 3s.

Again, forget the 0's.

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• 0

121121, 312132, 2341314 (Couldn't do 1,2 with just one pair)

Edited by CaptainEd
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• 0

23421314

Edited by kitnkat

• 0
23421314

4: Yes.

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• 0
121121, 312132, 2341314 (Couldn't do 1,2 with just one pair)

3: Yes.

4: Yes

2: - clever. but as you note, there is no solution for n=2.

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• 0

2--no

5--no

6--no

7, 61517346532472

What a fascinating kind of problem! An easy problem for backtracking, but I know I haven't the time to debug a program...

• 0

8642572468531713

• 0

7246258473651318

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2: None

3: 3,1,2,1,3,2

4: 4,1,3,1,2,4,3,2

5: None

6: None

7: 1,7,1,2,5,6,2,3,4,7,5,3,6,4

8: 1,7,1,2,8,6,2,3,5,7,4,3,6,8,5,4

9: None

10: None

11: 1,2,1,11,2,3,9,10,4,3,8,5,7,4,6,11,9,5,10,8,7,6

12: 1,2,1,3,2,12,10,3,11,4,5,9,6,8,4,7,5,10,12,6,11,9,8,7

13: None

14: None

15: 1,2,1,3,2,4,14,3,15,13,4,5,12,6,7,10,11,5,8,9,6,14,7,13,15,12,10,8,11,9

16: 1,2,1,3,2,4,16,3,13,5,4,15,12,14,6,5,7,8,11,9,10,6,13,16,7,12,8,15,14,9,11,10

It seems to me that there are only solutions for multiples of 4 and one less than multiples of 4.

class numbers
{
static boolean found = false;
static int size = 0;
static int numbers = 1;
static int[] thearray;
public static void main(String[] args)
{
while(true)
{
size = (++numbers)*2;
thearray = new int[size];
for(int i=0; i <size; i++) thearray[i] = 0;
found = false;
System.out.print("Searching for " + numbers + ": ");
findit(1);
}
}

public static void findit(int num)
{
for(int start = 0; start < size-(num+1); start++)
{
if ((thearray[start] == 0)&&(thearray[start+num+1]==0))
{
thearray[start] = num;
thearray[start+num+1] = num;

if (num != numbers) findit(num+1);
else
{
for(int i = 0; i < size; i++) System.out.print(thearray[i]+",");
System.out.println();
found = true;
}
if (found) return;

thearray[start] = 0;
thearray[start+num+1] = 0;
}
}
if (num==1) System.out.println("None");
}
}
`import java.util.*;`

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2: None

3: 3,1,2,1,3,2

4: 4,1,3,1,2,4,3,2

5: None

6: None

7: 1,7,1,2,5,6,2,3,4,7,5,3,6,4

8: 1,7,1,2,8,6,2,3,5,7,4,3,6,8,5,4

9: None

10: None

11: 1,2,1,11,2,3,9,10,4,3,8,5,7,4,6,11,9,5,10,8,7,6

12: 1,2,1,3,2,12,10,3,11,4,5,9,6,8,4,7,5,10,12,6,11,9,8,7

13: None

14: None

15: 1,2,1,3,2,4,14,3,15,13,4,5,12,6,7,10,11,5,8,9,6,14,7,13,15,12,10,8,11,9

16: 1,2,1,3,2,4,16,3,13,5,4,15,12,14,6,5,7,8,11,9,10,6,13,16,7,12,8,15,14,9,11,10

It seems to me that there are only solutions for multiples of 4 and one less than multiples of 4.

class numbers
{
static boolean found = false;
static int size = 0;
static int numbers = 1;
static int[] thearray;
public static void main(String[] args)
{
while(true)
{
size = (++numbers)*2;
thearray = new int[size];
for(int i=0; i <size; i++) thearray[i] = 0;
found = false;
System.out.print("Searching for " + numbers + ": ");
findit(1);
}
}

public static void findit(int num)
{
for(int start = 0; start < size-(num+1); start++)
{
if ((thearray[start] == 0)&&(thearray[start+num+1]==0))
{
thearray[start] = num;
thearray[start+num+1] = num;

if (num != numbers) findit(num+1);
else
{
for(int i = 0; i < size; i++) System.out.print(thearray[i]+",");
System.out.println();
found = true;
}
if (found) return;

thearray[start] = 0;
thearray[start+num+1] = 0;
}
}
if (num==1) System.out.println("None");
}
}
`import java.util.*;`

Searching for 2: None

Searching for 3: 0,0,3,1,2,1,3,2,

Searching for 4: 0,0,4,1,3,1,2,4,3,2,

Searching for 5: None

Searching for 6: None

Searching for 7: 0,0,1,7,1,2,5,6,2,3,4,7,5,3,6,4,

Searching for 8: 0,0,1,7,1,2,8,6,2,3,5,7,4,3,6,8,5,4,

Searching for 9: None

Searching for 10: None

Searching for 11: 0,0,1,2,1,11,2,3,9,10,4,3,8,5,7,4,6,11,9,5,10,8,7,6,

Searching for 12: 0,0,1,2,1,3,2,12,10,3,11,4,5,9,6,8,4,7,5,10,12,6,11,9,8,7,

Adding 0's does not make n*4+1 or n*4+2 possible...

class numbers
{
static boolean found = false;
static int size = 0;
static int numbers = 1;
static int[] thearray;
public static void main(String[] args)
{
while(true)
{
size = (++numbers)*2+2;
thearray = new int[size];
for(int i=0; i <size; i++) thearray[i] = -1;
found = false;
System.out.print("Searching for " + numbers + ": ");
findit(0);
}
}

public static void findit(int num)
{
for(int start = 0; start < size-(num+1); start++)
{
if ((thearray[start] == -1)&&(thearray[start+num+1]==-1))
{
thearray[start] = num;
thearray[start+num+1] = num;

if (num != numbers) findit(num+1);
else
{
for(int i = 0; i < size; i++) System.out.print(thearray[i]+",");
System.out.println();
found = true;
}
if (found) return;

thearray[start] = -1;
thearray[start+num+1] = -1;
}
}
if (num==0) System.out.println("None");
}
}
`import java.util.*;`

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• 0
2--no

5--no

6--no

7, 61517346532472

What a fascinating kind of problem! An easy problem for backtracking, but I know I haven't the time to debug a program...

5: Correct

6: Correct

7: Yes.

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It's a coloring argument. Take the array of numbers and color the cells alternating B and W.

1.1 is BWB, notice both 1's are B and the hole is W

121.2 is BWBWB, notice both 1's are B, the 2 is BW and the hole is W

312132 is BWBWBW, notice both 1's are W, both 3's are B, and 2 is BW

41312432 is BWBWBWBW, 1WW, 2BW, 3BB, 4BW

The odd numbers will always be BB or WW, the evens always BW.

4n will have 2n BW, and CAN have n BB and n WW (might not, in principle, but bear with me). If there are more than n Bs, then there will be some holes in W, and vice versa.

So, suppose you have a 4n that has no holes, because you were lucky and got n BB and n WW.

4n+1 will add BB (or WW, either way the argument works). Since there are two more Bs than Ws, there must be a W hole somewhere.

4n+2 adds a BW. Same story, there are still two more Bs than Ws.

4n+3 adds a BB or WW, so they MIGHT be able to balance BB with WW. If the number of Bs equal number of Ws, they match. I'm not proving they DO, just they CAN.

4n+4 adds another BW, which still permits the number of Bs to equal number of Ws.

So, while 4n+3 and 4n+4 CAN permit equal numbers of Bs and Ws, 4n+1 and 4n+2 CANNOT.

Thanks to EventHorizon for putting us out of our misery (manual backtracking got REAL OLD...) and showing us an interesting pattern, to boot.

Thanks to Bonanova for coming up with yet another interesting, non-trivial, and yet possible problem.

Edited by CaptainEd
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Nice proof CaptainEd! Wish I'd thought of it.

Here are solutions for leaving a single zero in (only when needed (n mod 4 == 1 or 2), though you could simply add in a zero to one end on the others). It seems they are all possible in this scenario.

Searching for 2: 1,2,1,0,2,

Searching for 3: 3,1,2,1,3,2,

Searching for 4: 4,1,3,1,2,4,3,2,

Searching for 5: 1,3,1,4,5,3,2,0,4,2,5,

Searching for 6: 1,3,1,4,6,3,5,2,4,0,2,6,5,

Searching for 7: 1,6,1,3,5,7,4,3,6,2,5,4,2,7,

Searching for 8: 1,3,1,7,5,3,8,6,4,2,5,7,2,4,6,8,

Searching for 9: 1,3,1,5,7,3,8,6,9,5,2,4,7,2,6,8,4,0,9,

Searching for 10: 1,3,1,5,7,3,9,10,6,5,2,8,7,2,4,6,9,0,10,4,8,

Searching for 11: 1,3,1,5,7,3,9,11,6,5,10,8,7,4,2,6,9,2,4,11,8,10,

Searching for 12: 1,3,1,5,7,3,11,9,12,5,4,10,7,6,8,4,2,9,11,2,6,12,10,8,

Searching for 13: 1,3,1,5,7,3,9,11,13,5,12,2,7,10,2,4,9,8,6,11,4,0,13,12,10,6,8,

Searching for 14: 1,3,1,5,7,3,9,11,13,5,14,2,7,12,2,8,9,10,4,11,6,0,13,4,8,14,12

,6,10,

Searching for 15: 1,3,1,5,7,3,9,11,13,5,12,15,7,10,14,2,9,8,2,11,4,6,13,12,10,4,

8,15,6,14,

Searching for 16: 1,3,1,5,7,3,9,11,13,5,14,15,7,16,8,12,9,2,10,11,2,4,13,8,6,14,

4,15,12,10,16,6,

Searching for 17: 1,3,1,5,7,3,9,11,13,5,15,17,7,16,2,8,9,2,14,11,10,12,13,4,8,6,

15,0,4,17,16,10,6,14,12,

Searching for 18: 1,3,1,5,7,3,9,11,13,5,15,17,7,18,2,14,9,2,16,11,6,12,13,4,8,10

,15,6,4,17,14,0,18,8,12,16,10,

Searching for 19: 1,3,1,5,7,3,9,11,13,5,15,17,7,19,16,14,9,2,18,11,2,12,13,4,10,

6,15,8,4,17,14,16,6,19,12,10,8,18,

Searching for 20: 1,3,1,5,7,3,9,11,13,5,15,17,7,19,20,16,9,2,18,11,2,12,13,14,6,

4,15,8,10,17,4,6,16,19,12,20,8,18,14,10,

Searching for 21: 1,3,1,5,7,3,9,11,13,5,15,17,7,19,21,2,9,20,2,11,16,18,13,8,14,

4,15,6,12,17,4,10,8,19,6,0,21,16,20,14,18,12,10,

Searching for 22: 1,3,1,5,7,3,9,11,13,5,15,17,7,19,21,2,9,22,2,11,18,20,13,14,16

,4,15,6,8,17,4,12,10,19,6,0,21,8,14,18,22,16,20,10,12,

Searching for 23: 1,3,1,5,7,3,9,11,13,5,15,17,7,19,21,23,9,2,22,11,2,16,13,20,18

,14,15,6,8,17,4,10,12,19,6,4,21,8,16,23,14,22,10,18,20,12,

Searching for 24: 1,3,1,5,7,3,9,11,13,5,15,17,7,19,21,23,9,2,22,11,2,24,13,18,20

,14,15,16,8,17,4,6,10,19,12,4,21,8,6,23,14,22,18,10,16,20,24,12,

Searching for 25: 1,3,1,5,7,3,9,11,13,5,15,17,7,19,21,23,9,2,25,11,2,20,13,24,22

,14,15,4,16,17,18,6,4,19,8,10,21,12,6,23,14,0,20,8,25,16,10,22,24,18,12,

Searching for 26: 1,3,1,5,7,3,9,11,13,5,15,17,7,19,21,23,9,2,25,11,2,26,13,22,18

,24,15,4,20,17,16,8,4,19,10,6,21,14,12,23,8,0,6,18,25,10,22,16,26,20,24,12,14,

Searching for 27: 1,3,1,5,7,3,9,11,13,5,15,17,7,19,21,23,9,25,27,11,2,26,13,2,20

,24,15,16,22,17,18,8,4,19,12,6,21,4,14,23,8,10,6,25,16,20,27,12,26,18,24,22,10,1

4,

Searching for 28: 1,3,1,5,7,3,9,11,13,5,15,17,7,19,21,23,9,25,27,11,2,28,13,2,26

,22,15,16,24,17,18,20,4,19,10,6,21,4,8,23,14,12,6,25,16,10,27,8,22,18,28,26,20,2

4,12,14,

Searching for 29: 1,3,1,5,7,3,9,11,13,5,15,17,7,19,21,23,9,25,27,11,29,2,13,28,2

,4,15,26,20,17,4,24,22,19,18,12,21,6,8,23,10,16,14,25,6,0,27,8,12,20,29,10,28,18

,26,22,24,14,16,

Searching for 30: 1,3,1,5,7,3,9,11,13,5,15,17,7,19,21,23,9,25,27,11,29,2,13,30,2

,4,15,28,24,17,4,26,22,19,14,16,21,10,20,23,6,18,8,25,12,0,27,6,10,14,29,8,16,24

,30,22,28,12,26,20,18,

Searching for 31: 1,3,1,5,7,3,9,11,13,5,15,17,7,19,21,23,9,25,27,11,29,31,13,28,

2,30,15,2,20,17,4,26,22,19,24,4,21,10,18,23,6,12,8,25,16,14,27,6,10,20,29,8,28,3

1,12,22,30,18,26,24,14,16,

Searching for 32: 1,3,1,5,7,3,9,11,13,5,15,17,7,19,21,23,9,25,27,11,29,31,13,32,

2,28,15,2,30,17,24,20,4,19,26,22,21,4,10,23,6,18,8,25,12,16,27,6,14,10,29,8,20,3

1,28,24,32,12,22,30,18,26,16,14,

Here's some slightly optimized code (force 1's to be on the left side or centered, and place odd numbers first and check parity (inspired by CaptainEd's proof)). If you don't allow the extra 0, you can force evenodd to be 0 instead of -1,0, or

1.

class numbers
{
static boolean found = false;
static int size = 0;
static int numbers = 1;
static int[] thearray;

static int evenodd;

public static void main(String[] args)
{
while(true)
{
size = (++numbers)*2;
if ((numbers%4 == 1)||(numbers%4 == 2)) size++;
thearray = new int[size];
for(int i=0; i <size; i++) thearray[i] = 0;
found = false;
System.out.print("Searching for " + numbers + ": ");
evenodd = 0;
findit0(1);
}
}

public static void findit0(int num)
{
int end = size-(num+1);
if (num == 1)
{
if (end > (size/2)+1) end = (size/2)+1;
}
for(int start = 0; start < end; start++)
{
if ((thearray[start] == 0)&&(thearray[start+num+1]==0))
{
thearray[start] = num;
thearray[start+num+1] = num;
if (start%2 == 0) evenodd++;
else evenodd--;

if (num+2 <= numbers) findit0(num+2);
else if (evenodd*evenodd < 4) findit(2);
if (found) return;

if (start%2 == 0) evenodd--;
else evenodd++;
thearray[start] = 0;
thearray[start+num+1] = 0;
}
}
if (num==1) System.out.println("None");
}

public static void findit(int num)
{
for(int start = 0; start < size-(num+1); start++)
{
if ((thearray[start] == 0)&&(thearray[start+num+1]==0))
{
thearray[start] = num;
thearray[start+num+1] = num;

if (num+2 <= numbers) findit(num+2);
else
{
for(int i = 0; i < size; i++) System.out.print(thearray[i]+",");
System.out.println();
found = true;
}
if (found) return;

thearray[start] = 0;
thearray[start+num+1] = 0;
}
}
}
}
`import java.util.*;`

• 0

1111?

• 0

for 1,2

121121 ?

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• 0
for 1,2

121121 ?

It works for the 2s but with all the 1s, not every pair spans just one digit.

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• 0

EventHorizon, you've given us reason to believe that 4n+3 and 4n+4 cases all have solutions, and that the others all have solutions with only one hole. Can you easily count how many solutions for the first few (like 7 and 8)? And if there's less than 100, could you print them out? (If this list looks too big, could you just PM them to me? I have tried thinking about how to prove that they exist and figure out an algorithm beyond backtracking, and can't get anywhere...)

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• 0
EventHorizon, you've given us reason to believe that 4n+3 and 4n+4 cases all have solutions, and that the others all have solutions with only one hole. Can you easily count how many solutions for the first few (like 7 and 8)? And if there's less than 100, could you print them out? (If this list looks too big, could you just PM them to me? I have tried thinking about how to prove that they exist and figure out an algorithm beyond backtracking, and can't get anywhere...)

I just happened to check brainden today. I modified my code to print out all the solutions (actually half of them, the rest are just reversed).

Here's the code:

class numbers
{
static boolean found = false;
static int size = 0;
static int numbers = 1;
static int[] thearray;

static int evenodd;

public static void main(String[] args)
{
while(true)
{
do
{
numbers++;
} while ((numbers%4 == 1)||(numbers%4 == 2));

size = (numbers)*2;
thearray = new int[size];
for(int i=0; i <size; i++) thearray[i] = 0;
found = false;
System.out.println("Searching for " + numbers + ": ");
evenodd = 0;
findit0(1);
System.out.println();
}
}

public static void findit0(int num)
{
int end = size-(num+1);
if (num == 1) end = numbers-1;
for(int start = 0; start < end; start++)
{
if ((thearray[start] == 0)&&(thearray[start+num+1]==0))
{
thearray[start] = num;
thearray[start+num+1] = num;
if (start%2 == 0) evenodd++;
else evenodd--;

if (num+2 <= numbers) findit0(num+2);
else if (evenodd*evenodd < 4) findit(2);

if (start%2 == 0) evenodd--;
else evenodd++;
thearray[start] = 0;
thearray[start+num+1] = 0;
}
}
}

public static void findit(int num)
{
for(int start = 0; start < size-(num+1); start++)
{
if ((thearray[start] == 0)&&(thearray[start+num+1]==0))
{
thearray[start] = num;
thearray[start+num+1] = num;

if (num+2 <= numbers) findit(num+2);
else
{
for(int i = 0; i < size; i++) System.out.print(thearray[i]+",");
System.out.println();
}

thearray[start] = 0;
thearray[start+num+1] = 0;
}
}
}
}
`import java.util.*;`

And here's its output (stopping at 8):

3,1,2,1,3,2,

Searching for 4:
4,1,3,1,2,4,3,2,

Searching for 7:
1,6,1,3,5,7,4,3,6,2,5,4,2,7,
1,5,1,7,3,4,6,5,3,2,4,7,2,6,
1,5,1,6,3,7,4,5,3,2,6,4,2,7,
1,4,1,6,7,3,4,5,2,3,6,2,7,5,
1,5,1,4,6,7,3,5,4,2,3,6,2,7,
1,7,1,2,5,6,2,3,4,7,5,3,6,4,
1,4,1,5,6,7,4,2,3,5,2,6,3,7,
1,7,1,2,6,4,2,5,3,7,4,6,3,5,
1,5,1,6,7,2,4,5,2,3,6,4,7,3,
1,6,1,7,2,4,5,2,6,3,4,7,5,3,
7,1,3,1,6,4,3,5,7,2,4,6,2,5,
6,1,5,1,7,3,4,6,5,3,2,4,7,2,
7,1,4,1,6,3,5,4,7,3,2,6,5,2,
4,1,6,1,7,4,3,5,2,6,3,2,7,5,
4,1,7,1,6,4,2,5,3,2,7,6,3,5,
5,1,7,1,6,2,5,4,2,3,7,6,4,3,
7,3,1,6,1,3,4,5,7,2,6,4,2,5,
5,6,1,7,1,3,5,4,6,3,2,7,4,2,
4,6,1,7,1,4,3,5,6,2,3,7,2,5,
5,7,1,4,1,6,5,3,4,7,2,3,6,2,
7,4,1,5,1,6,4,3,7,5,2,3,6,2,
4,6,1,7,1,4,5,2,6,3,2,7,5,3,
3,6,7,1,3,1,4,5,6,2,7,4,2,5,
5,7,4,1,6,1,5,4,3,7,2,6,3,2,
4,5,6,7,1,4,1,5,3,6,2,7,3,2,
2,6,7,2,1,5,1,4,6,3,7,5,4,3,

Searching for 8:
1,3,1,7,5,3,8,6,4,2,5,7,2,4,6,8,
1,3,1,8,5,3,6,7,2,4,5,2,8,6,4,7,
1,3,1,7,8,3,5,2,6,4,2,7,5,8,4,6,
1,3,1,6,8,3,5,7,2,4,6,2,5,8,4,7,
1,3,1,6,7,3,8,5,2,4,6,2,7,5,4,8,
1,3,1,6,8,3,4,7,5,2,6,4,2,8,5,7,
1,3,1,8,6,3,7,2,4,5,2,6,8,4,7,5,
1,3,1,6,8,3,7,4,2,5,6,2,4,8,7,5,
1,7,1,3,5,6,8,3,4,7,5,2,6,4,2,8,
1,6,1,3,7,5,8,3,6,4,2,5,7,2,4,8,
1,6,1,3,8,5,7,3,6,2,4,5,2,8,7,4,
1,7,1,3,8,4,5,3,6,7,4,2,5,8,2,6,
1,8,1,3,4,7,5,3,6,4,8,2,5,7,2,6,
1,6,1,3,7,8,4,3,6,5,2,4,7,2,8,5,
1,5,1,7,3,8,6,5,3,2,4,7,2,6,8,4,
1,5,1,7,3,6,8,5,3,4,2,7,6,2,4,8,
1,8,1,5,3,7,4,6,3,5,8,4,2,7,6,2,
1,7,1,6,3,8,4,5,3,7,6,4,2,5,8,2,
1,5,1,6,7,3,8,5,4,3,6,2,7,4,2,8,
1,8,1,4,6,3,7,5,4,3,8,6,2,5,7,2,
1,4,1,8,6,3,4,7,5,3,2,6,8,2,5,7,
1,5,1,8,4,7,3,5,6,4,3,2,8,7,2,6,
1,7,1,4,8,5,3,6,4,7,3,5,2,8,6,2,
1,7,1,4,6,8,3,5,4,7,3,6,2,5,8,2,
1,6,1,7,4,8,3,5,6,4,3,7,2,5,8,2,
1,4,1,5,7,8,4,3,6,5,2,3,7,2,8,6,
1,6,1,5,8,4,7,3,6,5,4,3,2,8,7,2,
1,7,1,4,5,8,6,3,4,7,5,3,2,6,8,2,
1,7,1,2,8,6,2,3,5,7,4,3,6,8,5,4,
1,5,1,6,4,7,8,5,3,4,6,2,3,7,2,8,
1,4,1,5,6,8,4,7,3,5,2,6,3,2,8,7,
1,7,1,2,8,5,2,6,3,7,4,5,3,8,6,4,
1,7,1,2,6,8,2,5,3,7,4,6,3,5,8,4,
1,5,1,8,6,2,7,5,2,3,4,6,8,3,7,4,
1,6,1,8,2,5,7,2,6,3,4,5,8,3,7,4,
1,6,1,7,2,8,5,2,6,3,4,7,5,3,8,4,
1,5,1,4,6,7,8,5,4,2,3,6,2,7,3,8,
1,4,1,5,8,6,4,7,2,5,3,2,6,8,3,7,
1,7,1,2,8,5,2,4,6,7,3,5,4,8,3,6,
1,6,1,8,2,7,4,2,6,5,3,4,8,7,3,5,
1,5,1,6,7,8,2,5,4,2,6,3,7,4,8,3,
1,8,1,5,2,6,7,2,4,5,8,3,6,4,7,3,
1,7,1,6,2,8,5,2,4,7,6,3,5,4,8,3,
1,7,1,8,2,4,6,2,5,7,4,3,8,6,5,3,
3,1,7,1,3,5,8,6,4,2,7,5,2,4,6,8,
3,1,8,1,3,7,5,2,6,4,2,8,5,7,4,6,
3,1,7,1,3,6,8,5,2,4,7,2,6,5,4,8,
3,1,7,1,3,8,4,5,6,2,7,4,2,5,8,6,
3,1,8,1,3,4,7,5,6,2,4,8,2,5,7,6,
3,1,8,1,3,4,6,7,5,2,4,8,2,6,5,7,
3,1,7,1,3,8,6,4,2,5,7,2,4,6,8,5,
3,1,8,1,3,6,7,2,4,5,2,8,6,4,7,5,
8,1,3,1,5,7,3,4,6,8,5,2,4,7,2,6,
8,1,3,1,5,6,3,7,4,8,5,2,6,4,2,7,
7,1,3,1,8,5,3,6,7,2,4,5,2,8,6,4,
7,1,3,1,6,8,3,5,7,2,4,6,2,5,8,4,
8,1,3,1,7,4,3,5,6,8,4,2,7,5,2,6,
7,1,3,1,6,8,3,4,7,5,2,6,4,2,8,5,
5,1,8,1,3,7,5,6,3,2,4,8,2,7,6,4,
5,1,8,1,3,6,5,7,3,4,2,8,6,2,4,7,
5,1,7,1,8,3,5,4,6,3,7,2,4,8,2,6,
6,1,8,1,5,3,7,6,4,3,5,8,2,4,7,2,
6,1,7,1,8,3,4,6,5,3,7,4,2,8,5,2,
8,1,4,1,6,7,3,4,5,8,3,6,2,7,5,2,
6,1,8,1,4,7,3,6,5,4,3,8,2,7,5,2,
7,1,4,1,8,6,3,4,7,5,3,2,6,8,2,5,
8,1,2,1,7,2,6,3,5,8,4,3,7,6,5,4,
4,1,6,1,7,4,8,3,5,6,2,3,7,2,5,8,
5,1,6,1,8,4,5,7,3,6,4,2,3,8,2,7,
8,1,2,1,6,2,7,5,3,8,4,6,3,5,7,4,
5,1,8,1,7,2,5,6,2,3,4,8,7,3,6,4,
6,1,5,1,7,4,8,6,5,3,4,2,7,3,2,8,
7,1,4,1,5,6,8,4,7,3,5,2,6,3,2,8,
8,1,2,1,6,2,5,7,4,8,3,6,5,4,3,7,
4,1,6,1,8,4,5,7,2,6,3,2,5,8,3,7,
8,1,2,1,7,2,4,5,6,8,3,4,7,5,3,6,
4,1,8,1,7,4,2,5,6,2,3,8,7,5,3,6,
4,1,6,1,7,4,8,5,2,6,3,2,7,5,3,8,
5,1,6,1,7,8,5,2,4,6,2,3,7,4,8,3,
6,1,5,1,8,4,7,6,5,2,4,3,2,8,7,3,
7,1,4,1,5,8,6,4,7,2,5,3,2,6,8,3,
6,1,7,1,8,2,5,6,2,4,7,3,5,8,4,3,
8,3,1,6,1,3,5,7,4,8,6,2,5,4,2,7,
7,3,1,8,1,3,4,6,7,5,2,4,8,2,6,5,
7,8,1,3,1,5,6,3,7,4,8,5,2,6,4,2,
8,6,1,3,1,7,5,3,6,8,4,2,5,7,2,4,
5,8,1,7,1,3,5,6,4,3,8,7,2,4,6,2,
7,5,1,8,1,3,6,5,7,3,4,2,8,6,2,4,
4,7,1,8,1,4,3,5,6,7,3,2,8,5,2,6,
5,7,1,6,1,8,5,3,4,7,6,3,2,4,8,2,
2,8,1,2,1,7,5,3,6,4,8,3,5,7,4,6,
8,4,1,6,1,7,4,3,5,8,6,3,2,7,5,2,
2,8,1,2,1,6,7,3,4,5,8,3,6,4,7,5,
4,6,1,8,1,4,7,3,6,5,2,3,8,2,7,5,
4,8,1,5,1,4,6,7,3,5,8,2,3,6,2,7,
2,8,1,2,1,5,6,7,3,4,8,5,3,6,4,7,
4,7,1,6,1,4,8,5,3,7,6,2,3,5,2,8,
6,4,1,7,1,8,4,6,3,5,2,7,3,2,8,5,
2,8,1,2,1,7,4,6,3,5,8,4,3,7,6,5,
5,8,1,4,1,6,5,7,4,3,8,2,6,3,2,7,
5,6,1,8,1,4,5,7,6,3,4,2,8,3,2,7,
7,5,1,6,1,8,4,5,7,3,6,4,2,3,8,2,
2,8,1,2,1,5,7,4,6,3,8,5,4,3,7,6,
6,8,1,4,1,5,7,6,4,3,8,5,2,3,7,2,
2,8,1,2,1,6,4,7,5,3,8,4,6,3,5,7,
4,6,1,7,1,4,8,5,6,2,3,7,2,5,3,8,
6,4,1,8,1,7,4,6,2,5,3,2,8,7,3,5,
5,6,1,8,1,7,5,2,6,4,2,3,8,7,4,3,
5,8,1,4,1,7,5,6,4,2,8,3,2,7,6,3,
8,5,1,4,1,6,7,5,4,8,2,3,6,2,7,3,
6,7,1,4,1,8,5,6,4,7,2,3,5,2,8,3,
7,8,1,2,1,6,2,5,7,4,8,3,6,5,4,3,
7,4,1,6,1,8,4,5,7,2,6,3,2,5,8,3,
4,7,1,8,1,4,6,2,5,7,2,3,8,6,5,3,
3,6,8,1,3,1,5,7,6,4,2,8,5,2,4,7,
3,8,6,1,3,1,5,7,4,6,8,2,5,4,2,7,
3,6,8,1,3,1,7,5,6,2,4,8,2,5,7,4,
3,6,8,1,3,1,7,4,6,5,2,8,4,2,7,5,
7,8,3,1,6,1,3,5,7,4,8,6,2,5,4,2,
8,6,3,1,7,1,3,5,6,8,4,2,7,5,2,4,
6,8,3,1,7,1,3,6,4,5,8,2,7,4,2,5,
5,6,7,1,8,1,5,3,6,4,7,3,2,8,4,2,
6,4,7,1,8,1,4,6,3,5,7,2,3,8,2,5,
5,8,4,1,7,1,5,4,6,3,8,2,7,3,2,6,
7,4,8,1,5,1,4,6,7,3,5,8,2,3,6,2,
7,2,8,1,2,1,5,6,7,3,4,8,5,3,6,4,
7,5,8,1,4,1,6,5,7,4,3,8,2,6,3,2,
7,5,6,1,8,1,4,5,7,6,3,4,2,8,3,2,
6,7,5,1,8,1,4,6,5,7,3,4,2,8,3,2,
7,2,8,1,2,1,6,4,7,5,3,8,4,6,3,5,
8,4,5,1,7,1,4,6,5,8,2,3,7,2,6,3,
8,2,7,1,2,1,5,6,4,8,7,3,5,4,6,3,
8,2,7,1,2,1,6,4,5,8,7,3,4,6,5,3,
6,4,7,1,8,1,4,6,5,2,7,3,2,8,5,3,
7,3,6,8,1,3,1,5,7,6,4,2,8,5,2,4,
7,3,8,6,1,3,1,5,7,4,6,8,2,5,4,2,
6,3,7,8,1,3,1,6,4,5,7,2,8,4,2,5,
4,5,7,8,1,4,1,5,6,3,7,2,8,3,2,6,
4,8,5,7,1,4,1,6,5,3,8,7,2,3,6,2,
2,6,8,2,1,7,1,4,6,5,3,8,4,7,3,5,
4,7,5,8,1,4,1,6,5,7,2,3,8,2,6,3,
2,8,6,2,1,7,1,4,5,6,8,3,4,7,5,3,
3,7,5,8,3,1,6,1,5,7,4,2,8,6,2,4,
3,8,5,7,3,1,6,1,5,4,8,7,2,6,4,2,
2,8,5,2,7,1,6,1,5,4,8,3,7,6,4,3,
2,6,7,2,8,1,5,1,6,4,7,3,5,8,4,3,
3,5,6,8,3,7,1,5,1,6,4,2,8,7,2,4,
3,5,8,6,3,7,1,5,1,4,6,8,2,7,4,2,
4,5,6,7,8,4,1,5,1,6,3,7,2,8,3,2,
2,5,7,2,8,6,1,5,1,4,7,3,6,8,4,3,
`Searching for 3:`

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