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The following article appeared in the June 2004 issue of the Canadian Paper Money Society Newsletter. It is an original work of the author, Brent W.J. Mackie. All rights reserved. © Copyright 2004 Brent W.J. Mackie.

Rotator Notes
Rarer Than 2-Digit Radar Notes But What Are They?

By: Brent W.J. Mackie
With contributions from: Michael A. La Croix

I was speaking with Michael La Croix, a mathematics combinatorics and optimization Ph.D. student at the University of Waterloo and a fellow currency collector, back in February of this year and he showed me a Birds series $5 note that he had in his collection. I looked at it for a moment, checked the serial number, examined it for errors, checked the condition, etc, and nothing significant stood out at me. It was close to a radar note, but not quite.

Then Michael turned the note upside-down so that the serial numbers were at the top. All of a sudden, it clicked. I could read the serial number without having to try reading backwards or twisting my head around! The serial number was symmetrical about the centre digit. The only difference between this type of number and a typical radar note is that this one is rotated about the centre of the middle digit, whereas in a radar note, the number is a reflection at the horizontal centre of the middle digit. I have decided to call this variety of notes "rotator notes", after the rotational symmetry of the serial number. It is also not a coincidence that the word "rotator" is symmetric.

This article will relate radars and rotators to mathematics, dive into the specifics of rotator notes, give some examples of them and wrap up by discussing the scarcity of rotator notes compared to that of radars.

The mathematics behind radar and rotator notes

Upper-year high school mathematics taught us about even and odd functions. An even function is one that is symmetrical on both side of the vertical axis. An odd function is one that is symmetrical about the point (0, 0). Good examples of even and odd functions are illustrated in Figure 1 and Figure 2 below.

Figure 1

Figure 1: An even function: f(x) = x2

Figure 2

Figure 2: An odd function: f(x) = sin(x)

As you can see from the graphs, if you flip the even function f(x) = x2 along the vertical axis (ie: x = 0), you will get the same image. Further, if you rotate the odd function f(x) = sin(x) 180° about the origin (0, 0), you will again get the same image. Notice though, that if you rotate the even function or flip the odd function, the result is a different image than what you started with.

Radar notes vs. rotators: the specifics

Relating this back to serial numbers, radar notes are easily compared with even functions. When you have a 5 on the right, you've got a 5 on the left, the same distance from the centre digit. Refer to Figure 3 for an example of a radar note. From the outside of the serial number towards the centre digit, both sides of the 7-digit number are the same. There are no restrictions on which numbers can be used on a radar note. With rotator notes, it's a little bit trickier.

Figure 3

Figure 3: A standard 4-digit radar note

Rotator notes are the equivalent to an odd function in that you must get the same number if you rotate it by 180°. The only digits that will allow this directly are 0 and 8. If you rotate them around, they're the exact same digit. Obviously, you cannot do this with a 3, 4, 7, etc. When you rotate the digit 6, it becomes a 9, and vice versa. Note that on all Bank of Canada note series, the digit 1 has at least one serif on it at the top. Some incarnations of it (namely the 1954 and earlier notes) include a significant serif along the bottom of the digit. Thus, it cannot be considered able to rotate. Because 0 and 8 are the only digits that rotate to themselves, they are the only ones that can be used as the centre digit of the serial number. The outer 3 sets of digits form an "odd function", if you will. This function will accept any digit from 0, 6, 8 or 9 and return the rotated digit. So if you have a 6 in the first position, you must have a 9 in the last position.

Examples of rotator notes

Here are some examples of rotator notes: FNX 8888888 (also a solid radar), HNA 6680899, EJA 9690696, ESD 0008000 (again a radar), CBI 9088806, EKA 9860986 (a "repeater") and FMD 6668999. Refer to Figure 4 for a scan of a rotator note serial number and Figure 5, the rotated serial number. Note that as with radar notes, the alphabetic prefix of the serial number is not taken into account when considering rotator notes.

Figure 4

Figure 4: A rotator note

Figure 5

Figure 5: A rotator note, rotated

Scarcity of radar and rotator notes

As mentioned earlier, there are 4 choices for each of the first three digits of the serial number, namely 0, 6, 8 and 9. There are only two choices for the centre digit, namely 0 or 8. Then, the last 3 digits are predetermined based on the first 3 digits. Therefore, there are 4x4x4x2 = 128 possibilities for rotator serial numbers. Subtract 0000000 (which is destroyed) and you're left with 127 rotator notes per 10,000,000 notes printed. (There are only 126 from BA Banknote since 0008000 is also removed and destroyed).

In comparison, there are only 630 (10x9x7) two-digit radar notes per 10,000,000 notes printed. You have 10 choices for the first digit, 9 choices for the second digit and there are 7 distinct ways to arrange those digits. Without going into a mathematical derivation, assume that there are also 5,040 4-digit radars (10x9x8x7) and 4,320 3-digit radars (10x9x8x6) and 9 solid radars for a total of 9,999 radars per 10,000,000 notes. (Again, BA Banknote will reduce this number to about 9,990 radars because it destroys the entire sheet containing 0000000 and 9 other radars).

Collecting rotator notes

Before being shown the rotator note depicted above (HNA 6680899), never before have I heard of anyone collecting such a thing. Upon further investigation though, rotator notes seem to be slightly less unknown south of the border. In fact, in American numismatic circles, rotator notes are referred to as "swims" notes, since the word "swims" itself bears the same properties of the notes it describes. As well, these notes were briefly discussed as a small part of a larger article in an issue of the IBNS journal in 2000. Radar notes have long been collectible and approximate values are listed in the Charlton Standard Catalogue of Canadian Government Paper Money. I believe that given enough publicity and time, rotator notes will become sought-after among collectors of Canadian paper money.