Mystic Number Machine

The  “The Henry Ford” collection[1]  contains a unique Burroughs adding machine: The Mystic Number Machine (figure 1) This printing adding machine has only 8 keys: a “Memory Bar”, a “Magic Key” and 6 keys with a strange symbol. The instructions on the machine read:


HERE'S HOW
Pick Any Number from 1 to 60.
Locate the Magic Squares in Which Your Number Appears.
On Each Square is a Mystic Symbol. Find This Symbol on the Keyboard of the Mystic Number Machine, and Depress That Key. Then Depress the Memory Bar.
Repeat this Procedure — Pressing First the Proper Mystic Symbol and Then the Memory Bar — Until You Have Depressed all the Keys for the Squares in Which Your Number Appears.
Then Depress the Magic Key. The Machine Will Print the Number You Have in Mind.
(If You Depress a Wrong Symbol Key, Press the Magic Key and Start Over) 

The Henry Ford indicates that the machine was made by a Burroughs employee around 1950 as a one-off, but a Burroughs machine with the same properties is described in the catalog of the 1933–1934 Chicago World Fair.[2]

The machine has serial number A977708. Burroughs started around 1934 using serial numbers beginning with an “A”[3]  and stopped using them around 1950. A high serial number would be expected at the end of that period. The machine also looks modern, with a hood covering the ribbon spools and a logo[4]  introduced around 1947. The location of the serial number at the bottom of the front[5]  also indicates that the machine was made around 1950, as The Henry Ford states. But then another machine was used during the Chicago World Fair of 1933–1934. Where did that machine go?

Burroughs logo since ca. 1947

The instructions remind me of a children's magic tricks box I used to own, which contained a set of cards you could use to “guess” a number a member of the audience was thinking of. The cards looked like this:

The trick is simple if you think of binary representations of the numbers: each card contains numbers that share a specific bit. The first number of each cards indicates this bit: 1, 2, 4, 8, 16, 32. By asking the audience member which cards contain the chosen number, and secretly the adding the first numbers of these cards, you get the chosen number.

But these cards are not squares, let alone magic squares.

The Henry Ford does not own the Magic Squares that are mentioned in the instructions. What would these magic squares have looked like? We can try to split each card with 32 numbers from  the magic trick box into two  cards with 4×4=16 numbers. Half of these cards will show the number-to-be-added (1, 2, 4, 8, 16, 32), but the other half will not. That's no problem  because they are marked with the Mystic Symbol that represents that number.  For each large card, the split has to result in two magic squares: square grids filled with unique numbers and giving, per square, the same sum for all rows, columns and diagonals.

Magic squares

A simple way to create magic squares is described in the 18th century by Leonhard Euler, but the method is much older:[6] 

Choose 4 different numbers a, b, c, d and 4 different numbers α, β, γ en δ. Put 4 a's, b's, c's and d's in a square grid in such a way that each row, column and diagonal contains a certain letter only once. Do the same with 4 α's, β's, γ's and δ's and add the two squares. The sum of each row, column or diagonal in the resulting square is a+b+c+d+α+β+γ+δ. There is no guarantee that the numbers in the resulting square are all different. That requires some restrictions on the numerical values of a, b, c, d and α, β, γ, δ.

When splitting the large cards  into Magic Squares for the Mystic Number Machine the same method can be applied.  Again, it is convenient to use a binary representation of the numbers.

As an example we construct the two cards for the Mystic Number 1, so having the least significant bit (20) equal to 1. 

For a, b, c en d we use numbers with 20=1 and all possible values for bits 21 and 22, so 1, 3, 5 and 7. For α, β, γ and δ we use numbers with 20=0 and all possible values for bits 23 and 24, so 0, 8, 16 and 24. In the final magic square the numbers a, b, c and d are never added to each other, so the numbers in the magic square will all have bit 20 equaling 1. Because α, β, γ and δ are not added to each other as well, no  “two's carry” will take place during the construction of the magic square. Therefore all numbers in the square are unique and the square is a real, normal, magic square.

For the second card  for Mystic Number 1  we use the same values for a, b, c and d, but for α, β, γ and δ set bit 25 to 1.
This results in a Magic Square with numbers larger than 32.


In the construction of Magic Squares for the remaining Mystic Numbers a similar recipe is used: set the bit corresponding to the Mystic Number to 1 in a, b, c and d and to 0 in α, β, γ and δ; use the remaining least-significant bits for a, b, c, d and use the remaining higher bits for α, β, γ, δ, with bit 25 equal to 0 for one card and  25 equal to 1 for the other. For Mystic Number 32=25  bit 24 could be used to distinguish between one card and the other.


Machine construction

The machine is probably based on a Burroughs Portable adding machine with a full keyboard, i.e. a keyboard with multiple columns with keys 1 to 9. The keys with Mystic Symbols press one or two keys of that full keyboard. The Memory Bar adds the numbers that are entered. The Magic Key prints the sum and resets the machine to 0. The Memory Bar would not have been necessary if a Comptometer-like machine had been used: numbers could be added immediately. Burroughs also made Comptometers, but no printing Comptometers. Felt & Tarrant, the maker of the original Comptometer, did make one: the Comptograph.

If we assume that the linkages between the Mystic Keys and the  (invisible) full keyboard should be as short as possible, the most likely order of the keys, from the front of the machine, is: 1, 2, 32, 4, 16, 8.
This also explains the spacing of the keys:

Conclusion

The complete set of Magic Squares could have looked like this:

The disadvantage of the Magic Squares compared with the large cards from the magic tricks box is that the numbers are not ordered. This increases the risk of the audience overlooking a number on a Magic Square, and then the trick will fail. An advantage is that the trick is less obvious: the relation between the numbers on a Magic Square is not easily seen. The manual prescribes 60 as the maximum number. Strictly speaking it should be 63, but that could give a hint about the underlying trick. The Magic Squares can also be used with mental arithmetic, so without the special adding machine: just add the first numbers of the matching  Magic Squares, but remember to subtract 32 from the first number if it exceeds 32.

A Dutch version of this paper has appeared in MIR 84,  february 2021