Kinda Cool Numbers

[LArdennais]

UUabbcbbcbbb

[lmviterbo]

UUabbcbbcbbb

Aaaaargh! Just one note away from a 2-digit radar on the same sheet!

[g.s.51]

What do you think of the following note ?
5€ VCcggcggcggc ?
An extremely rare palindrome!

[lmviterbo]

What do you think of the following note ?
5€ VCcggcggcggc ?
An extremely rare palindrome!

That one is quite something!!!

It's more or less the 10-digit equivalent to what RB Edgar & Alan A. Moss call an Oscillating Radar (on their The Green Guide of Fancy Serial Numbers : A Beginners Guide). The book focuses on dollar banknotes and their 8-digit serial numbers. Finding an equivalent kind of fancy serial on a 10-digit serial is often much less probable.

You can find the "Oscillating Radar" on page 176, coded CS-1330. Examples given are 44144144 and 22522522. They grade its rarity 8, on a scale of 0 – 10 (mind you, on an 8-digit serial). And they estimate its OFV (Over Face Value) to be $1,500 – $10K+.

Although I don't know the market for fancy serials in Euro banknotes (that sometimes show up on Delcampe and other platforms), I suppose that the same Indian, American and Chinese fancy serial collectors who are used to trade and collect 8-digit fancy serials should pay attention to the much rarer 10-digit ones.

[g.s.51]

OPEN AI estimates that there is a 1 in 7.4 million chance (0.0000135%) of encountering a palindrome in a 10-digit check number, given that the first and last digits are not 0.
This is therefore an extremely rare occurrence !

[lmviterbo]

OPEN AI estimates that there is a 1 in 7.4 million chance (0.0000135%) of encountering a palindrome in a 10-digit check number, given that the first and last digits are not 0.
This is therefore an extremely rare occurrence !

Now that is an extremely wrong answer by OpenAI.

Total 10-Digit Numbers where the last digit can't be a zero (but the first one can be) is exactly the same total as if the first digit couldn't be a zero (but the last one can). That is all numbers from 1,000,000,000 to 9,999,999,999. This is also the same as 1,000,000,001 to 10,000,000,000, that is, exactly 9,000,000,000.

Total Palindromes: 90,000 (Formed by choosing the first 5 digits, where the first digit cannot be 0: 9 x 10 x 10 x 10 x 10 = 90,000.

Probability Formula: 90,000 / 9,000,000,000 = 1 in 100,000, or 0.001%.

This is confirmed by real stats too.

[g.s.51]

It is surprising to note that OPEN AI’s reliability is questionable when it comes to probabilities !?
That should be its strong point !
It is generally accepted that the probability of finding just two numbers in a 10-digit lottery draw is approximately 1 in 217,000.
One might legitimately be surprised that a palindrome (a rarity among rarities) consisting of just two digits in a 10-digit check number has a probability of 1 in 100,000.
In other words, the palindromic rarity (1 in 100,000) is therefore less rare than the basic rarity (1 in 217,000).
Now that’s a rather irrational paradox...

[lmviterbo]

I reckon it's a perception paradox. If you look at a glance at any 10-digit serial number, you will rarely miss a "binary" (only two different digits used), but you will easily miss a "radar" (a palindrome), unless it's a binary or a trinary.

[g.s.51]

Probabilities are not a matter of opinion, or a paradox exception, but a mathematical proof that is supposed to be irrefutable.
An exceptional distribution of digits within a 10-digit check digit (in this case, a palindrome consisting of just two digits) cannot be any less rare than a check digit with two digits in random order.....

[Tárrega]

I got this one in in February. Didn't save it.. Hopefully it wasn't too valuable, it was in good condition, and I let it go..

[lmviterbo]

Ah, @g.s.51, I see now exactly where our wires got crossed! We are comparing apples to oranges.

When you quoted OpenAI's figure of 1 in 7.4 million (0.0000135%), that is actually the correct probability for a binary palindrome (like the gorgeous VCcggcggcggc you posted earlier).

However, in your post, you just used the word "palindrome," so I provided the probability for a general palindrome / general radar (which can use any mix of digits, like 1845225481).

You are 100% correct in your logic: a palindrome restricted to only two digits is a rarity among rarities, and it is exponentially harder to find than a regular binary note. The paradox disappears once we separate standard radars from binary radars.

To sum up the math for valid 10-digit serials:

• Standard Binary (any order): ~1 in 217,000
• General Radar (any digits, mirrored): 1 in 100,000
• Binary Radar (only two digits, mirrored): ~1 in 7,400,000

________________

As to Tárrega's 9-digit radar, it's a very cool find but I believe it's not that rare. I haven't confirmed the following odds but I think they are correct (for a 9-digit binary radar on any of the possible positions within a serial number): ~1 in 24,200 notes on Europa banknotes; or ~1 in 16,700 notes on 2002 notes.

[Tárrega]

Oh, I'm happy to hear that! Next time I'll take it slightly more easy, so I don't sent something potentially rare away..

[LArdennais]

Found yesterday : NNaabaccccca

[philgelico]

Found today :