# Plate Position Arithmetic

The mathematical relationship between a note's serial number and its plate position can be used to detect a counterfeit or altered note, and is sometimes also helpful in determining the length of a print run when BEP records are missing or incorrect. The calculations themselves are not complex, but the details have changed several times over the years with changes in the BEP's serialling technology, so it is important to apply the correct rule for any given note.

If you'd rather not do all the work yourself, here's an Excel sheet that carries out the calculations described below.

### Short skip numbering (LEPE), 2012-present

The BEP's new Large Examining and Printing Equipment (LEPE) applies serial numbers to the sheets in a different pattern than any previous technology. LEPE has been used with both 50-subject sheets and 32-subject sheets.

To calculate the correct plate position for a LEPE note, first take the serial number and subtract 0.5. Divide the result by either 5000 (for 50-subject printings) or 3200 (for 32-subject printings); and then subtract off the whole-number part of the result, leaving only the decimal. Multiply this decimal by 50 or 32, as appropriate, and round up to the next whole number. Then convert the result to a plate position using the appropriate chart:
50-subject sheet
41 = A1 31 = A2 21 = A3 11 = A4 1 = A5
42 = B1 32 = B2 22 = B3 12 = B4 2 = B5
43 = C1 33 = C2 23 = C3 13 = C4 3 = C5
44 = D1 34 = D2 24 = D3 14 = D4 4 = D5
45 = E1 35 = E2 25 = E3 15 = E4 5 = E5
46 = F1 36 = F2 26 = F3 16 = F4 6 = F5
47 = G1 37 = G2 27 = G3 17 = G4  7 = G5
48 = H1 38 = H2 28 = H3 18 = H4 8 = H5
49 = I1 39 = I2 29 = I3 19 = I4 9 = I5
50 = J1 40 = J2 30 = J3 20 = J4 10 = J5
32-subject sheet
25 = A1 17 = E1 9 = A3 1 = E3
26 = B1 18 = F1 10 = B3 2 = F3
27 = C1 19 = G1 11 = C3  3 = G3
28 = D1 20 = H1 12 = D3 4 = H3
29 = A2 21 = E2 13 = A4 5 = E4
30 = B2 22 = F2 14 = B4 6 = F4
31 = C2 23 = G2 15 = C4 7 = G4
32 = D2 24 = H2 16 = D4 8 = H4
Notice that, for either sheet size, LEPE always numbers from right to left, top to bottom. The 50-subject plate positions are laid out gridwise; the 32-subject plate positions are laid out by quadrants, because they were designed to follow the COPE numbering (see below). Also notice that under the LEPE system, the plate position depends only upon the serial number, not upon the print run size; this makes the calculation simpler, but also means that we can no longer determine unknown run sizes by looking at plate positions.

Example: Consider 2013 \$1 K11223344A. This note was printed on LEPE in a 50-subject sheet. So we compute:

• 11223344 - 0.5 = 11223343.5
• 11223343.5 / 5000 = 2244.6687
• 2244.6687 - 2244 = 0.6687
• 0.6687 * 50 = 33.435
• 33.435 rounded up is 34
And so by the 50-subject-sheet chart, this note should have plate position D2.

### Long skip numbering (COPE & predecessors), 1952-present

This numbering scheme has been used for all 18-subject sheets and all 32-subject sheets except those printed on LEPE. Before calculating the plate position corresponding to a given serial number, it is necessary to know the size of the print run used; the table of standard print run sizes may be helpful.

To calculate a plate position, first determine the run size in notes, by multiplying the number of sheets in the run by 32 or 18, as appropriate. Take the serial number, subtract 0.5, and divide by the number of notes in the run. Subtract off the whole-number part of the result, leaving only the decimal. Multiply this decimal by 32 or 18, and round up to the next whole number. Then convert the result to a plate position using the appropriate chart:
32-subject sheet
1 = A1 5 = E1 17 = A3 21 = E3
2 = B1 6 = F1 18 = B3 22 = F3
3 = C1 7 = G1 19 = C3 23 = G3
4 = D1 8 = H1 20 = D3 24 = H3
9 = A2 13 = E2 25 = A4 29 = E4
10 = B2 14 = F2 26 = B4 30 = F4
11 = C2 15 = G2 27 = C4 31 = G4
12 = D2 16 = H2 28 = D4 32 = H4
18-subject sheet
1 = A  7 = G 13 = M
2 = B 8 = H 14 = N
3 = C 9 = I 15 = O
4 = D 10 = J 16 = P
5 = E 11 = K 17 = Q
6 = F 12 = L 18 = R
Keep in mind that COPE actually cuts the sheets in half vertically before serialling them, so it is possible (though somewhat uncommon) for the halves to be swapped, resulting in a note whose plate position is on the "wrong" half of the sheet--A3 instead of A1, for example. No similar swaps occur on 32-subject sheets printed before COPE was introduced (in 1971), or on 18-subject sheets.

Example: Consider 1950C \$20 E87654321B. This note was printed in an 18-subject sheet, in a run of 20,000 sheets (or 360,000 notes). So we compute:

• 87654321 - 0.5 = 87654320.5
• 87654320.5 / 360000 = 243.484224
• 243.484224 - 243 = 0.484224
• 0.484224 * 18 = 8.716025
• 8.716025 rounded up is 9
And so by the chart, this note should have plate position I.

The above calculation will fail for certain notes printed in irregular runs, such as the gap-free partial star runs printed since 1995 and many of the uncut-sheet runs printed since 1981. In these cases, it is necessary to look up not only the number of notes in the run, but also the starting serial of the run--here the group lists for the various series may be helpful. Then, take the serial number of the note, subtract the starting serial number of its run, and add 0.5. Divide by the number of notes in the run, multiply by 32, round up to the next whole number, and convert to a plate position using the chart above.

Example: Consider 2004A \$10 GL03333333*. This note was printed in a short run of 512,000 star notes (16,000 sheets) beginning at serial GL03200001*. So we compute:

• 3333333 - 3200001 + 0.5 = 133332.5
• 133332.5 / 512000 = 0.260415
• 0.260415 * 32 = 8.333281
• 8.333281 rounded up is 9
And so by the chart, this note should have plate position A2--though A4 is also a possibility if a COPE swap has occurred.

(Note: A few very small special printings, such as the Millennium stars and some of the Lucky Money issues, used extremely nonstandard sheet layouts that break even this calculation. For those notes, just look up the plate position on the charts on this page.)

Because the plate position appropriate to any given serial number is dependent on the size of the print run, we can often work backward from observed notes to determine the run size. This method has been used to produce many of the estimated serial ranges on this website, especially for star runs that were not reported, or incorrectly reported, by the BEP.

### Consecutive numbering, 1861-1953

Nearly all notes printed in 12-subject sheets, and nearly all large-size notes, used this numbering pattern, in which serial numbers run sequentially down each sheet. In cases where the sheet has more than one column (the small-size 12-subject sheets and large-size 8-subject sheets), the columns are used independently of one another and may be cut apart before numbering, so it is not possible to calculate which side of a sheet a given serial number will come from, only its position within that side. Thus the 12-subject sheets are thought of as sheets of 6 notes, while the large-size 8-subject and 4-subject sheets are all thought of as sheets of 4 notes. A few rare series of high-denomination large-size currency were printed in sheets of 3 notes.

To calculate a plate position, simply determine the remainder when the serial number is divided by 6, 4, or 3, as appropriate. (That is, divide the serial number by 6, 4, or 3; subtract off the whole-number part of the answer, leaving only the decimal; and multiply by 6, 4, or 3 again.) Then convert the result to a plate position using the appropriate chart:
12-subject sheet
1 = A or G
2 = B or H
3 = C or I
4 = D or J
5 = E or K
0 = F or L
8- or 4-subj. sheet
1 = A or E
2 = B or F
3 = C or G
0 = D or H
3-subj. sheet
1 = A
2 = B
0 = C
Here again, the plate positions are independent of the print run size. During the era when consecutive numbering was in use, run sizes were not standardized; notes could be printed in runs of whatever length was convenient. For 12-subject notes, print runs can sometimes be identified by the BEP's tendency to print the first half of a run on the A-F side of the sheets and the second half on the G-L side, but various exceptions to this tendency do exist.

### Sheet numbering, 1863-1933

This numbering pattern, in which all notes on a sheet receive the same serial number, was used only for National Bank Notes (except the 1929 Type II issues, which used consecutive numbering as above). Obviously, under such a system, any serial number can be paired with any plate position--indeed, both together are necessary in order to identify an individual note.

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