The Case of the Miscalculating Calculator

Question

At the Arithmetic Company, to try and cut down labor costs the administration commissioned a calculator to be made. The calculator was produced, and in the quality control division was tested with the following values:

20 + 3 = 23
5 + 15 = 20
7 + 9 + 9 = 25
28 - 22 + 2 = 8
33 - 20 + 4 = 17
71 + 5 - 75 = 1
30 + 19 - 5 = 44
45 + 36 - 23 = 58

Finding all the results satisfactory they put the calculator to use. The first client requested various sums of 2, and using the calculator, was given the following list:

2 + 2 = 85
2 + 2 + 2 = 150
2 + 2 + 2 + 2 = 152
2 + 2 + 2 + 2 + 2 = 232
2 + 2 + 2 + 2 + 2 + 2 = 1114
2 + 2 + 2 + 2 + 2 + 2 + 2 = 1167
2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 2227
2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 17727
2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 24727

The next client requests some consecutive sequences and was given:

1 + 2 + 3 + 4 = 130
5 + 6 + 7 + 8 = 130
9 + 10 + 11 + 12 = 188
13 + 14 + 15 + 16 = 1216
17 + 18 + 19 + 20 = 1157

Finally the last client requests just one sum and is given:

1 + 1 = ERROR

The next day they received many complaints from their clients, and decide to bring in you, their best analyst, to figure out what the calculator is doing. Your job is to find the rules that the determine the calculator's output as well as one example sum (not listed above) so that your method can be tested.

Bonus: What does the calculator output when given 1×1 ?

---

People may ask any number of additional sums to the calculator in the chat room http://chat.stackexchange.com/rooms/22378/sum-requests:

0 + 0 = ERROR
1 + 2 = ERROR
1 - 3 = ERROR
2 - 7 = ERROR
3 + 3 = ERROR
4 + 4 = ERROR
5 + 19 = ERROR
19 + 5 = ERROR
76 - 75 = ERROR
130 - 85 = ERROR
1 - 2 + 3 + 4 = ERROR
2 + 2 - 2 = 2
15 = 7
15 + 5 = 20
18 = 18
19 = 19
7 + 40 = 25
29 = 25
13 + 4 = 26
39 = 35
5 + 5 = 40
9 + 9 = 40
40 = 40
49 - 5 = 40
5 + 5 + 5 = 45
49 = 45
59 = 55
6 + 2 = 70
5 + 5 + 5 + 5 = 72
85 = 85
3 + 7 = 91
2 + 85 = 150
85 + 2 = 150
85 + 85 = 152
71 + 5 = 303
45 + 13 = 517

---

If necessary, I'll provide a new hint for few days the problem remains unsolved.

Hint 1: When inputting solitary numbers into the calculator you get the following results:

0 = 0
1 = 1
2 = 2
3 = 3
4 = 4
5 = 5
6 = 6
7 = 7
8 = 8
9 = 5
10 = 10

Hint 2:

55 + 1 = 35
59 + 1 = 35
65 + 1 = 107
69 + 1 = 107
75 + 1 = 303
79 + 1 = 303

Hint 3:

15 + 3 = 91
19 + 3 = 35

Answer 1

It took over a year, but it is solved!

The calculator works by converting each number into its english representation (1 --> "one"), and then converting the word back into a number using the system of English Gematria. Operators such as plus and minus work normally. The answer to each equation is the smallest number whose Gematria value matches the left hand side of the equation, or ERROR if no such number exists.

Example. 2 + 2 = 85 because "two" = 348 and "eighty-five" = 696 which is 348 + 348.

Example. 1 + 1 = ERROR because "one" = 204 and there is no number whose Gematria value is 408.

Here are some example Gematria values:

zero       635
one        204
two        348
three      336
four       360
five       252
six        312
seven      390
eight      300
nine       252 = five
fifteen    390 = seven
nineteen   516
twenty     642
forty      504 = five + five
eightyfive 696 = two + two

The answer to the bonus, 1x1, is the number whose Gematria value is equal to 204 * 204 = 41616. This would be a staggeringly huge number and I can't calculate it unless I make a program for it. Maybe someone else can?

EDIT: As suggested in the comments, the intended solution is to convert each letter into a number value (a=1, b=2, ...). Gematria is the same thing except the values are multiplied by 6, so (a=6, b=12, ...). For addition and subtraction, there is no difference.

But for the bonus, the intended solution is 1x1 = "one" x "one" = 34 x 34 = a number whose letter value is 1156, where a=1, b=2, etc. Still a large number, but much more solvable.

Answer 2

NOT AN ANSWER

This is not intended to be an answer. The content is simply too involved for a comment. This may all be useless.

---

First, here's a little VBA UDF you can pop into Excel to have it evaluate the formula. If you plug in the whole formula as an equality such as they are given in the examples, it'll return TRUE or FALSE. If you plug in just one side of the formula, it'll return the solution.

Public Function eval(s As Range) As String
    eval = Application.Evaluate(s.Value)
End Function

Put it in a module and call it in a cell with a formula like =eval(A1). This is probably the only piece of this posting that'll be helpful.

---

Regarding the Sums of Twos

The answers for the examples that are all sums of twos show a pattern when plotted on a logarithmic scale.

---

Regarding the 9=5 Example

As of the time of this posting, there are 72 example formulas given.

24 / 72 evaluate as true on a normal calculator.

28 / 72 evaluate as true if you replace all the 9s with 5s

There are 6 that are true originally but become false if you replace the 9s and 2 that are true but become false.

The examples in Hint 2 become 3 unique examples with each repeated twice if you replace the 9s with 5s.

The calculator may or may not interpret every 9 keypress as a 5

The only answers with a 9 in them are 19=19 and 3 + 7 = 91

---

Regarding the ERROR Values

I think the key is in the errors. What kind of logic could dictate that 0+0=ERR and 1+1=ERR? It's not just a simple operator substitution here. There are several more examples of the format A+B=ERR and I have yet to find something that fits them. I feel like there's something key here but I can't grasp it.

---

Answer 3

For the bonus question, we can determine that we do not need to use the word "and" in our conversions to text, as it's given that 2 + 85 = 150, and this requires 150 to be spelled "one hundred fifty".

Running a program to find a number that evaluates to "one" * "one" = 34 * 34 = 1156 gives 1,121,272,777, or "one billion one hundred twenty-one million two hundred seventy-two thousand seven hundred seventy-seven".

If we did require the word "and" in the numbers, the answer to 1 * 1 would be 222,274,777, or "two hundred and twenty-two million two hundred and seventy-four thousand seven hundred and seventy-seven".