During the week 18-23, July 2021, contributions came from Giorgos
Kalogeropoulos, Emmanuel Vantiegehm, Simon Cavegn
I believe that the example with 11 is unique.
If we consider the more
relaxed case of integers and calculate the difference of the binary
and its reverse we will see that 11 is the only case where the
difference equals 2.
I also noticed a rule
for which I don't have proof:
The minimum difference
between a binary and its reverse of an integer k with 2^n < k <
2^(n+2) (if we exclude zero which is the case of binary
This means that the
minimum difference >0 for numbers greater than 2^4 is 2^2=4. So if
we are searching for numbers whose difference is 2 we should only
look from 2^2 up to 2^4=16.
If we do this we will
find that the only number with this property is 11.,
Another example to make
this rule clearer is that for integers k with 2^10 < k < 2^12 the
minimum difference >0 of the binary and its reverse is 2^5 = 32
The only numbers with
difference 32 are 2081, 2225, 2345, 2489, 2597, 2741, 2861, 3005,
3107, 3251, 3371, 3515, 3623, 3767, 3887, 4031 and they are smaller
The answer to Puzzle 1046 is
: there are no further examples.
There are even no more odd numbers
whose binary form reverses when 2 is added.
Let's work binary.
Then we state :
The only number that reverses by adding 10 is 1011.
It is clear that we must restrict to odd digits.
First of all, the number 1011 after adding 10 becomes 1101, the
reverse of 1011.
We see immediately (by exhausting all possibilities) that there
is no other four- (or less) digit number with that property.
Take a five( or more) digit number m. Then there are four
m = 10...01. Then : m+10 = 10...11, not the reverse of m
m = 11...01 : it's reverse is 10...11, smaller than m, thus
certainly not equal to m+10
m = 11...11. Adding 10 ends in 01, certainly not the
reverse of m.
m = 10...11. Then : m+10 must be 11...01. which can be
obtained only if the intermediate digits of m are 1.
But adding 10 will turn them into zeros, which is not
giving the reverse !