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Pandigitals Divided by 11 (Posted on 2017-07-23) Difficulty: 3 of 5
For which bases b is it possible to make a pandigital number in base b that is divisible by 11 in base b?

For example, in base 4: 1023 / 11 = 33. (In base 10 this is 75 / 5 = 15.)

No Solution Yet Submitted by Brian Smith    
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Solution Solution by cases Comment 4 of 4 |
As was pointed out by Larry, the base Mod 4 is important.  Also, the base 10 divisibility rules for 11 at true in other bases.

The sum of the digits in base B is S=B(B-1)/2.  This sum can be even or odd for a base that is even or odd.

Case 1: B=0mod4
the base is even and the sum is even.  The sum can easily be partitioned into two equal set of the same size and formed into the pandigital number.
here's an easy way to do it with an example base 8
0 3 4 7
1 2 5 6 
snake up and down, each square of four numbers has two upper numbers sum the same as the two below.  Riffle these together to get a number: 10235467
It is obvious how to extend this.

Case 2: B=1mod4
base is odd but sum is even.  
similar to Case 1 but leave out the 0.  Here's base 9
1 4 5 8
2 3 6 7
Riffle together, put the zero at the end 
124356870

Case 3: B=2mod4
base is even but sum is odd.  The digits cannot be put into two equal sets, but they can be put into sets that differ by B+1.
Here's how in base 10:
2 4 5 8 9 (sum=28)
1 3 6 7 0 (sum=17) (28-17=11)
Riffle to get 1234658790
[Note this doesn't work for base 2 because the 1 would have to go the the top and then end of the bottom.]
***edit:  I haven't fully shown how this generalizes but the idea is to put the largest number opposite the 0 and flip other pairs to add/subtract 1***

Case 4: B=3mod4
base is odd and sum is odd.  The digits cannot be put into two equal sets.  Making them differ by B+1 (or any multiple of B+1) is not possible either.
Example in base 11.  S=55
Let the sums of the sets be x and y.
x+y=55
x-y=12
the solutions are not integers.  Making the 12 become 24 doesn't help...

Edited on July 23, 2017, 8:49 pm
  Posted by Jer on 2017-07-23 16:34:08

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