If you're thumbing through your change looking for cool serial numbers, you may run across a bill that has eight consecutive digits but jumbled up. Something like 17456832.
A term that has gained some traction is “mixed ladder” — being like a ladder serial number but mixed up. (Ladder serial numbers have the digits in order, like 23456789.) This kind of bill can also be called a “scrambled ladder.”
What are the odds of finding a mixed ladder serial number?
(WARNING: Math ahead!)
Bills entering circulation in recent years have serial numbers 00000001 to 96000000.
There are four sequences of consecutive digits: 0 to 7, 1 to 8, 2 to 9, and 3 to 0 (taking 0 to be like a 10).
For each of these sequences, there are 8! (8 factorial) ways to arrange them. Eight factorial is 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320. Multiply by the four sequences and we get 161,280 ways.
We have to subtract a few though. First, we take out all serials that start with 98, 97, and 96, because they will be greater than 96,000,000. For each of 98, 97, and 96, there are 6! = 720 ways to arrange the remaining digits. Both sequences 2 to 9 and 3 to 0 have a 9 in them. This makes 3 x 2 x 720 = 4,320 sequences we have to take out because they exceed 96 million.
Since we're dealing with mixed ladders and not regular ladders, we'll need to take them out too. 98765432 was taken out in the step above, so there are 7 regular ladders left to take out: four ascending ladders, and three more descending ladders.
This gives us the population of mixed ladders per run as 161,280 – 4,320 – 7 = 156,953 mixed ladders.
Finally, we take this as a fraction of the whole population: 156,963 / 96,000,000 = 0.0016392, or about 1 in 612 bills.
What are scrambled ladder serial numbers worth?
The answer to this question always depends on who you ask. Many currency dealers will consider these bills to be worth face value, and some even go so far as to say that there's no such thing as a mixed ladder.
With one out of 612 bills being a mixed ladder, they're not particularly rare, but at the same time, they're not particularly easy to find, either. If you hunt currency for serial numbers, it's likely that you'll find one with time.
Condition, age, and other factors also affect what a bill may be worth to a collector. If we consider only the serial number being a mixed ladder on a $1 bill, this is usually worth between $2 and $4 over face value on eBay. (The range of values from several listings I found was between $2.75 and $12.99. The one that went for $12.99 was also a star note.)
Should you buy a mixed ladder serial number?
Since mixed ladders aren't particularly rare, you're bound to find one simply by checking your change or withdrawing a few hundred $1s from the bank to go hunting. Then it costs you face value (plus time, but I find it enjoyable!)
If you really want an example and don't want to wait to find one, then there's of course nothing wrong with buying one. I would aim to buy one for $4 shipped at the most.
I would advise against buying a whole lot of them because it would be unlikely that you would recoup your money. If you take them to a dealer, it would be nearly impossible to recoup your money. But buying one for $4 is cheap entertainment and a neat conversation piece.
Can you sell a mixed ladder serial number?
If you can buy one, you certainly can sell one.
You'll want to sell it for a price that makes money, though.
If selling on eBay, you'll want to consider the listing fee (if any), the final value fee, merchant fees (the fee for accepting a payment by credit card), and shipping fees (probably an envelope, a few index cards, and a forever stamp). Plus, of course, the cost of the note!
If you can sell a $1 bill for $5 shipped and mail it in an envelope with some protection inside for the bill, you'll likely make a small profit.
Is selling a scrambled ladder serial number scammy?
A fair number of currency dealers will call selling these kinds of bills scammy.
The premise they use is that these bills are not worth more than face value, so trying to sell them for more than face value is ripping people off.
To me, this reflects more of an individual decision not to buy and sell them than it does a lack of value. As a potential seller, I would just go elsewhere to sell them. And, based on seeing eBay sales happen regularly for these kinds of bills, the statement that they're not worth more than face value is simply false based on the evidence.
Photo by Lance Grandahl on Unsplash
