Plate position arithmetic

Okay, you folks asked for it.... Here's how to calculate plate positions on modern U.S. notes. It's going to be math-heavy; don't say I didn't warn you....

First of all, to avoid some complications, let's assume the note is (a) dated 1990 or later, (b) denominated $1 through $20, and (c) not a star note, uncut sheet note, or similar. (I'll cover these other cases later.)

Currency is printed in sheets of 32 notes, and in print runs of 200,000 sheets (=6,400,000 notes). The notes on each sheet are *not* consecutive; rather, consecutive notes come from the same position on different sheets. That way, the BEP can take a stack of 100 sheets, run the whole stack through a guillotine cutter, and cut it directly into a bunch of ready-to-package 100-note straps--there's no need to sort and stack the notes after they've been cut.

The serial numbering in each plate position runs consecutively through the entire 200,000-sheet print run. So in the first run of a block (for example), serial numbers 00000001 through 00200000 will be printed in position A1 of the 200,000 sheets. Serials 00200001 through 00400000 will be printed in position B1, and so on, until serials 06200001 through 06400000 are printed in position H4. Then the cycle starts over again in the next print run. A total of 15 of these print runs brings the numbering up to 96000000, at which point there aren't enough serials left for another whole run, so the numbering then moves on to the next block. (This is why serials above 96000000 aren't used these days.)

The upshot of this is that, if you look at all the notes in serial order, the plate position only changes once every 200,000 serials, and it cycles through the 32 possible positions once every 6,400,000 serials. So to calculate the correct plate position for a given serial number, you do the following:

1. Divide the serial number by 6,400,000. Discard the quotient and keep only the remainder.
2. Divide that remainder by 200,000, and round the answer *up* to a whole number. (Note that you *always* round up, even if the decimal is less than 1/2.)
3. You now have a number from 1 to 32, which corresponds to the plate position. Numbers 1 to 8 are positions A1 to H1; numbers 9 to 16 are positions A2 to H2; numbers 17 to 24 are positions A3 to H3; and numbers 25 to 32 are positions A4 to H4.

For example, if the serial number is 87654321, then dividing by 6,400,000 gives 13 with a remainder of 4454321. Dividing 4454321 by 200,000 gives about 22.27, which rounds up to 23. Thus this serial number belongs in position G3.

If you've got a calculator that doesn't have a divide-with-remainder function, then the steps can be restated this way: Divide the serial by 6,400,000; subtract the whole-number part of the answer, leaving only the decimal; multiply by 32; and round up to a whole number. So for example 87654321 divided by 6,400,000 is 13.695988; subtracting off the 13 leaves 0.695988; and multiplying by 32 gives about 22.27, which rounds up to 23, as above.

There's one potential complication still: the BEP actually cuts the 32-note sheets in half vertically, into 16-note pieces, before running them through the numbering presses. It's possible, though somewhat uncommon, for a half-sheet to end up in the "wrong" pile--a left half on the right side of the press, or a right half on the left side. When this happens, the plate position will be off by 16 places, meaning that the number 1 might be swapped for 3, or 2 might be swapped for 4. (So our example serial number 87654321 *should* have position G3, as calculated above, but *might* have position G1 instead.) Swaps like this aren't considered errors, since the BEP doesn't really mind them, but as a percentage of all notes they're pretty uncommon.

$50's, $100's, and older notes

Okay, the above discussion is great for anything printed in 6,400,000-note standard runs. What about other run sizes? Current-series $50's and $100's, as well as lower-denomination notes printed from about 1984 to 1989, used runs of 3,200,000 notes (=100,000 sheets) instead. The effect on the calculations is that you need to replace 6,400,000 by 3,200,000, and replace 200,000 by 100,000. Everything else still works just the same.

So if that serial number 87654321 were found on a $50 note, or on a 1985 $1, then the calculation would look like this: 87654321 divided by 3,200,000 is 27.391975; subtracting off the 27 leaves 0.391975; and multiplying by 32 gives 12.54, which rounds up to 13. So this note should have plate position E2 (but might be E4 instead).

(Incidentally, when there are only 3,200,000 notes per print run, the BEP can fit 31 print runs per block, leading to a maximum serial number of 99200000. That's why $50's and $100's use higher serials than the other denominations do, and why serials higher than 96000000 can be found on older notes of all denominations.)

Even earlier, print runs of various other, shorter lengths were used. And before Series 1957, the sheet size was different too: there were 18 notes per sheet (with the 18 positions labelled with single letters A through R). Possible run sizes over the years have included 1,280,000 notes (=40,000 sheets of 32); 640,000 notes (=20,000 sheets of 32); 256,000 notes (=8000 sheets of 32); 360,000 notes (=20,000 sheets of 18); and 144,000 notes (=8000 sheets of 18). The changes from one run size to another happened at different times for different denominations and types of currency, often in the middle of a series. So if you want to calculate a plate position on an older note, you'll first need to determine the correct print run standard from the mostly-complete table here.

Once you've done that, the calculation proceeds much as above: Divide the serial number by the number of notes in a standard print run; subtract off the whole-number part of the answer, leaving only the decimal; multiply by the number of notes printed per sheet; round *up* to a whole number; and convert that number to a plate position.

So for one more example, let's suppose that we've got a 1953 $5 silver certificate with serial number A87654321A. It's dated before 1957, so it was printed on an 18-note sheet, and the table shows that the appropriate standard print run is 20,000 sheets, or 360,000 notes. So we divide 87654321 by 360,000, obtaining 243.48423; subtract off the 243, leaving 0.48423; multiply by 18, obtaining about 8.72; and round up to 9. Thus this note comes from the 9th position on the sheet, position I. (And the 18-note sheets weren't cut before numbering, so I is the only possibility.)

If you go back a few more years, before about 1952, everything was printed in sheets of 12 notes, which were numbered in a completely different way. So none of the above will apply to notes printed in sheets of 12.

Star notes, uncut sheet notes, &c.

Incredibly, these are even more complicated. I'll come back to them when I've got time to make another gimungous post....

Meanwhile, if you're getting sick of doing all this arithmetic, here's an Excel sheet I rigged up that'll do the work for you.

 

Log in or Sign up to hide this ad.

All I can really say is - WOW !
Thanks alot for the very informative post.

 

Makes logical sense to me...thanks
Appreciate the explanation..now I know more than I did a few min ago!

RickieB

 

If you have Excel (or just use Google Documents), you can plug this formula into any cell other than B2 and then put your serial number in cell B2 and it will tell you the plate position.

=CHAR(CODE("A")+INT(MOD(MOD(B2-1,6400000)/200000,8)))&INT((MOD(B2-1,6400000)/200000)/8)+1

Little sloppy, but I wanted to get it all in one line. Need to change the 6400000 and 20000 if using anything other than the 32 note sheets.

Tested it with the notes in my wallet and it seems to work.

Oops...didn't see your spreadsheet. Like your solution much better!

 

very excellent Numbers thanks from the very informative post. you make it all too easy to look at a note and make sure everythign on it 'jives'

 

Thanks, now I see why you are called Numbers.

 

wow that was very informative! my question is....how did you figure all that out?

 

Some currency books explain how the printing/block numbering works. This site also has some good info:
http://www.uspapermoney.info/general/number.html

Once you know how the notes are printed, Numbers just applied some mathematical calculations to determine the plate position. He did a pretty good job of explaining the math part of it in the original post.

 

Went back and fixed a small error in my one line Excel calculation and after staring at it for a while, came up with an alternate way to calculate the plate position which is a little shorter.

1) Subtract 1 from serial number
2) Divide result from step 1 by number of sheets per run and round down.
3) Divide result from step 2 by number of notes printed per sheet and take the remainder.

This should give you the offset within the plate of the note from 0 to 31. Using the prior example.

1) 87654321 - 1 = 87654320
2) 87654320 / 200000 = 438.2716 rounded down to 438
3) 438.2716 / 32 = 13, remainder 22 is offset to note within plate.

To convert the 22 to the actual letter/number pair divide the result from step 8. Take the remainder and add to letter "A" to get the letter and then take the quotient + 1 to get the number.

Example:

22 / 8 = 6, remainder 2. So, "A"+6 = "G", 2+1 = 3, Plate position is G3.

Haven't tested it with all numbers yet, but seems to produce accurate results. In a single cell Excel formula, would look like this:

=CHAR(CODE("A")+MOD(MOD(INT((B2-1)/200000),32),8))&INT(MOD(INT((B2-1)/200000),32)/8+1)

Not too pretty and does some of the work twice. If you break it down the steps though it becomes clearer. In Excel, you could put in the following values:

B2 = The serial number of note
B3 = B2 - 1
B4 = INT( B2 / 200000 )
B5 = MOD( B4 , 32 )
B6 = CHAR( CODE( "A" ) + MOD( B5 , 8) ) & INT( B5 / 8 ) + 1

Note: The MOD() function gives remainder. Also, if your version of Excel or Google documents supports the QUOTIENT() function, the cells B4 and B6 could be re-written as:

B4 = QUOTIENT( B2 , 200000 )
B6 = CHAR( CODE( "A" ) + MOD( B5 , 8) ) & QUOTIENT( B5 , 8 ) + 1

 

nice dursin.. but numbers way seems a bit easier

and Numbers is the site owner, operator, admin, and maintanance man for www.uspapermoney.info

 

I guess I just like playing with numbers too...must be bored today! Was trying to understand the math behind it and started playing around with alternate ways to come up with the same answer. You know, I totally missed Numbers calculator shortcut...that makes it really easy. For some reason though, that one gets a bit ugly when you try to put it in terms Excel understands.

Didn't know that he was the owner of www.uspapermoney.info. GREAT site. I actually mirror it onto my harddrive just in case it goes away unexpectedly like mycurrencycollection.com did. Too valuable a resource to lose! Great work Numbers!!!

 

yeah it's just really neat...bet it took a lot of work getting the numbers right! yeah the original post was very informative as well

 

About those star notes....

Aw, shucks.... Glad you find it useful. And yes, it's too bad about mycurrencycollection.com vanishing the way it did; that star note lookup tool was pretty neat. Someday when I have much more free time, I'd like to learn enough about whatever-it-is-that's-required that I could put together webpages that do lookups that way, or that duplicate the function of the Excel sheet I posted, or other such things. But at the moment I don't really know anything about web construction beyond how to type up excessively oversized HTML tables.

Anyway, speaking of star notes...let the math-nerdery continue!

As long as the BEP is printing full runs of star notes, the plate position calculations work just like they do on regular notes. The only quirk is that the standard run size for stars hasn't always been the same as for regular notes; nowadays, for example, stars of all denominations are still printed in runs of 100,000 sheets (=3,200,000 notes), even though regular notes of most denominations are printed in runs twice that size. All that means is that when you look up the standard run size in the table, you have to make sure you're looking at the star table and not the regular table. Easy enough.

The reason there gets to be a lot more added complexity in the star notes is that the BEP often prints star notes in partial print runs--sometimes because they don't need a full 3.2 million stars at once, and other times because an odd-length star run is a convenient way of using up whatever printed sheets happen to be left over at the end of the printing of a series. Worse still, the BEP's numbering conventions for these partial star runs have varied quite a bit over time. Let me go over these different conventions in order from simplest to most complex to explain, rather than in chronological order....

From about 1977 to 1995, partial star runs were implemented by simply setting up the numbering press exactly as though a full star run were being printed, and then stopping the press part way through the run. Therefore, if you're just trying to check whether your star note has the correct plate position, you don't even have to care about whether it comes from a full run or a partial run; just look up the standard run size and do the calculation exactly as described in the top post. (On the other hand, if you're trying to figure out how many star notes were printed for a given series, then you definitely have to keep track of the partial runs, since a side effect of this approach to numbering is that every partial run ends up having a bunch of gaps in the serial ranges printed, rather than neatly using all the serials in between the lowest number printed and the highest number printed.)

Before 1977...some even odder things were going on, which created gaps of other types or in other places within the star runs. But the good news is, the 1977-1995 rule still applies: none of these odd things were allowed to affect the plate position of a note, so the method described above will still give you the correct plate positions.

From about 1999 to the present, a different numbering scheme is used--one which has the advantage of not allowing serialling gaps within print runs, but which unfortunately messes with the plate positions. Under this system:

(a) Every print run of star notes gets a full 3,200,000 serials assigned to it, even if it isn't going to use them all. Thus every run begins at a serial number that's one more than a multiple of 3,200,000. Run #1 for a given denomination and district always begins at 00000001, run #2 at 03200001, run #3 at 06400001, and so forth.

(b) When a partial star run is printed, the serials actually used come from the low end of the run's assigned range. So for example if run #2 is going to comprise only 60,000 sheets (=1,920,000 notes) instead of the full 100,000 sheets, then it'll use serials 03200001 to 05120000. Serials 05120001 to 06400000 will be assigned but unused; run #3 will still start at 06400001 when it comes along. Thus numbering gaps can occur *between* runs, just not *within* them.

(c) The skip between serials on each sheet is equal to the number of sheets actually printed in the run. So in the example above, the first 60,000 serials (03200001 to 03260000) will come from position A1, the next 60,000 serials (03260001 to 03320000) from position B1, and so forth. This is pretty much what you'd expect for the numbering of a 60,000-sheet print run. The trouble is that the run's starting serial was chosen based on the 100,000-sheet standard rather than the 60,000-sheet actual run, which is enough to break the formulas we were using.

To correct for this, we "adjust" our note's serial number to what it *would* have been if the run had started at 00000001. So if our note is actually from run #2, we need to subtract 3,200,000 from the serial number before doing the rest of the steps outlined in the top post--and when we do those steps, we use the actual run size of 1,920,000 notes, not the standard run size.

For example, let's say we've got a 2003 $100 DB04040404*. Checking the production figures, we see that this note was printed in a run exactly as described above: 1,920,000 notes long, starting at serial 03200001. So, we subtract 3,200,000 from the serial, obtaining 840,404. We divide by the run size of 1,920,000, obtaining about 0.4377. This already has a whole-number part of 0, so we go on to the next step of multiplying by 32, obtaining about 14.01, which rounds up to 15. Thus this note should have plate position G2 (but G4 is possible).

Actually, the division step in the above process will always produce a number less than 1, because of the extra subtraction at the beginning. So we could streamline the calculation: Instead of dividing by 1,920,000 and then immediately multiplying by 32, we could simply divide by 60,000, the run length expressed in *sheets* rather than notes.

So our rule is: Take the serial number, subtract the starting serial of the print run, divide by the number of sheets in the print run, round *up* to a whole number, and convert that number to a plate position.

One more example: Suppose we have 2003A $1 C06543210*. The production stats show that this one came from a particularly odd partial run that used serials 06400001 through 07008000; that's a run of 608,000 notes, or 19,000 sheets. So we calculate 06543210 - 06400000 = 143210, and then divide by 19,000, getting about 7.54, which rounds up to 8. Thus this note should have plate position H1 (or perhaps H3).

From about 1995 to 1999, everything in the previous section still applies, with one exception. During these years, the BEP would occasionally break a star run into two sub-runs. For purposes of the plate position calculation, you have to consider *only* the sub-run your note is actually in. And if it's the second of the two sub-runs, then the starting serial of that sub-run may be a good bit odder looking than a straight multiple of 3,200,000. But use it in the formula anyway, and all will be well.

For example, consider 1995 $1 A10509130*. Just by looking at the number, you can tell that this note is from star run #4 (which would cover serials 09600001 to 12800000, or part of that range). But the production reports indicate that run #4 of the 1995 $1 Boston stars was printed as two sub-runs: first A09600001* to A10496000*, and then A10496001* to A10880000*. (Serials 10880001 to 12800000 were thus not used at all.) Our note is from the second sub-run, and that run comprises 10880000 - 10496000 = 384,000 notes, or 12,000 sheets. So, we subtract the 10496000 from the serial number, obtaining 13,130; we divide this by the 12,000 sheets, obtaining about 1.09; and we round this up to 2. Thus our note should have plate position B1 (or perhaps B3).

Uncut sheets are similar. You need to know how long the run was (which is easy--it's the difference between the serials of the notes on the sheet) and the starting serial for the run. Then the method described for the recent star notes will work just fine. Unfortunately, the BEP doesn't generally tell us the starting/ending serials for the sheet runs--we end up working them out ourselves, by collecting data on several sheets from a given printing, and then using all of the above calculations in reverse to work out the original print run.

(Actually, that works for star notes, too--we've caught a few errors in the BEP production reports by realizing that the the star notes from a certain print run didn't actually have the plate positions that the BEP said they should've had. So, if you find a star note with a wonky plate position, report it and see if you've discovered another case like this....)

Finally, let me point out that the Excel sheet I linked in the top post *is* also set up to handle the irregular print runs discussed in this post...just give it the starting serial for the run in the appropriate cell, along with the other data. Hope this helps!

 

Wow...lot to read there! Better take it slow or my head will explode...or implode...either way, ain't gonna be pretty!

Shouldn't be too hard to do these calculations on a web page. Should just be some simple scripting. I'm just now getting into web programming at work after spending the past 14 years as a C++ developer so I can't help yet.

Anyway, thanks again for all the work you put into this stuff.

btw: The reference sections of mycurrencycollection.com are back up as of today. He even has a link back to your site as one of the 3 links on his page. Now, off to mirror that site before he changes his mind and takes it back down!

 

Numbers...do you have any information on how to calculate the plate position for notes printed on 12-subject sheets?

 

Those are actually a lot simpler to calculate, since the serialling system was quite different.... The sheets were cut in half vertically before numbering, and then the six notes on each half-sheet were numbered consecutively from top to bottom. So all you have to do is divide the serial number by 6 and look at the remainder: if it's 1, the note should be from position A or G; if it's 2, B or H; and so on.

There's no way to determine from the serial number whether a note will be from the left half of the sheet (A-B-C-D-E-F) or the right half (G-H-I-J-K-L), but it's also not completely random. If you look at consecutive runs of serial numbers, the next note after an F is nearly always an A, and the next note after an L is nearly always a G. So the BEP seems to have printed a bunch of consecutive numbers on left half-sheets, and then another bunch of consecutive numbers on right half-sheets, rather than mixing them. Unfortunately these bunches didn't come in standard sizes; near as we can tell, they may've been based on however many notes the BEP was planning to print that day, or some such thing.

The few surviving 12-subject uncut sheets typically have 12 consecutive serial numbers on them, with the numbering proceeding straight from position F to position G. Thus these sheets appear to have been handled specially and not numbered in the usual way. Some exceptional sheets (the Hawaii and North Africa $1's) have six consecutive numbers on each side with a gap of several thousand serials between the two sides; these are presumably representative of the BEP's usual numbering at the time.

It should also be noted that the 1929 Type 1 Nationals were serialled differently, using sheet numbers, not note numbers. That is, on any given half-sheet, all six notes would have the *same* serial number (just with a different prefix letter on each). So obviously it's possible for any number to fall in any plate position on these notes.... But the Type 2 Nationals switched to the same system as all other currency, so the divide-by-6 rule works for them.

Incidentally, this calculation works for nearly all large-size notes too, except with sheets of 4 rather than 6. Typically the plate positions were just A-B-C-D, though toward the end of the large-size era the BEP started using some 8-subject plates...the resulting sheets were cut in half vertically and then numbered in the same way as the 4-subject sheets, so positions E-F-G-H are equivalent to A-B-C-D respectively. Once again, though, Nationals were the exception; all large-size Nationals used sheet numbers rather than note numbers, so that the serial number and plate position are independent of one another. And a few early Gold Certificates were printed in sheets of 3 rather than 4, but those are pretty rare to run across....

Hope this helps!

 

Thanks, that does help a lot. Was hoping to add a 3rd tab to your spreadsheet, but looks like it's not worth it if it can't be made accurate due to not being able to tell which side the note was printed on.

btw: You should add a "Plate Position Arithmetic" page to your website.

Went out and looked at examples of uncut sheets from closed Heritage auctions and just as you stated, all the serials run in sequence for the sheet of 12 with the exception of the Hawaii and North Africa $1 sheets. I looked at 4 of them and the right side serial numbers were always 1998 greater than the left which seems like it indicates they were printing batches of 333 sheets.

 

Hmmm...actually, that's a good idea. Bits of it are here and here, but I guess I don't have a page that really goes into detail on all the math. I suppose I figured that anybody who'd be able to understand all the calculations involved would also be able to work it all out on their own--y'know, the way you posted about six variations of the formulas higher up this thread. But you're right; it never hurts to have it all compiled together in one place somewhere.

Great, now I've got *another* entry on my to-do list....

 

Great thread Numbers!

Thanks ~ Darryl

 

If you look at consecutive runs of serial numbers, the next note after an F is nearly always an A, and the next note after an L is nearly always a G. So the BEP seems to have printed a bunch of consecutive numbers on left half-sheets, and then another bunch of consecutive numbers on right half-sheets, rather than mixing them. Unfortunately these bunches didn't come in standard sizes; near as we can tell, they may've been based on however many notes the BEP was planning to print that day, or some such thing.

Numbers got it right, except I would like to clarify one thing:

Actually, 12-subject sheets were printed as the whole sheet and were then cut into 6-subject half-sheets after the application of serial numbers. Here, the presses were programmed to print the lower and upper half of the entire assigned serial number range on the left and right half-sheets, respectively. The reason that you always find F-A and L-G position sequences (such as in changeover pairs) is that the resulting half-sheet piles ended up in different bricks after cutting and wrapping.