# Explain: 5-1=1, 5+1=1, 5=1

Explain the following equations. The final answer is 1 word.

$$4 ÷ 7 - 1 = 0\\ 1 \times 12 + 8 = 2\\ 0 ÷ 10 + 7 = 0\\ 9 \times 2 - 3 = 1\\ 5 - 1 = 1\\ 4 - 2 = 1\\ 2 \times 2 - 9 = 1\\ 5 = 1\\ 5 + 1 = 1\\ 6 ÷ 4 - 1 = 0\\ 3 \times 5 = 1\\ 5 - 1 = 1\\ 4 ÷ 2 - 4 = 0\\ 3 - 2 = 0\\ 1 \times 15 + 5 = 2\\ 0 ÷ 4 = 0\\ 5 \times 3 - 1 = 1$$

the answer is right there, on second thought I have removed the tag, because there isn't really much to calculate.

• @newQOpenWid that's two words. Aug 11 at 22:45
• This isn't too related to the actual question, but, y'know, if you start with a wrong assumption, you can derive anything. Aug 13 at 14:48
• Also, why was 5-1 = 1 repeated? Is that deliberate? Aug 13 at 14:50
• V'ir abgvprq gung rirel pbzovangvba bs nevguzrgvp bcrengbef pbeerfcbaqf gb gur fnzr erfhyg. Guhf, V fhfcrpg gur ahzoref gb gur yrsg bs gur rdhnyf fvta ner zrnavatyrff. Ubjrire, V unira'g orra noyr gb svther bhg jung gb qb jvgu guvf. Fvzcyr fbyhgvbaf (r.t., ercynpvat ahzoref gb gur yrsg bs gur rdhnyf fvta jvgu bar) qb abg jbex. Aug 17 at 18:42
• @Brian Not quite true, actually: 5-1=1 and 3-2=0...
– Stiv
Aug 19 at 7:21

I think the answer may be...

...either CONCATENATION, LOOP, or the hyphenated CONCATENATION-LOOP.

This can be found by considering that...

...there are an awful lot of '0' and '1' values after the equals signs, and the only other number which appears (albeit less so) is '2'. These are all leading digits for numbers in the range 01-26, suggesting that there may be an alphabetical code in play, reliant on A1Z26.

With this in mind, if we consider...

...the digits that follow the equals sign, and pair them with the first digit of the calculation on the next line (note that the first number on each line is always a single digit itself), we can extract the following numbers:

01, 20, 09, 15, 14, 12, 15, 15, 16, 03, 15, 14, 03, 01, 20, 05, 14 (wrapping around from the last line to the first)

In A1Z26, these numbers convert to the letters:

A T I O N L O O P C O N C A T E N

And it doesn't take much now to realise that this spells out CONCATENATION LOOP when starting at the first 'C'.

There's also a way built in to the puzzle to confirm that this is the answer - the calculations are not merely window dressing, because:

If we apply the calculation that immediately follows the A1Z26 numbers, we produce the next one! Here, it's probably helpful to move the leading digit from each line to the end of the one before:

$$÷ 7 - 1 = 01$$
$$\times 12 + 8 = 20$$
$$÷ 10 + 7 = 09$$
$$\times 2 - 3 = 15$$
$$- 1 = 14$$
$$- 2 = 12$$
$$\times 2 - 9 = 15$$
$$= 15$$
$$+ 1 = 16$$
$$÷ 4 - 1 = 03$$
$$\times 5 = 15$$
$$- 1 = 14$$
$$÷ 2 - 4 = 03$$
$$- 2 = 01$$
$$\times 15 + 5 = 20$$
$$÷ 4 = 05$$
$$\times 3 - 1 = 14$$

In other words, each calculation serves as an instruction how to generate the next letter in the sequence from the previous one (when considering both numerically), looping back from the last line to the first, thereby living out its 'loop'.