# How many scouts are there?

At camp every tent is given a number. My twin lives at the reverse of my tent number. The difference between our tent numbers ends in two. What are the lowest possible numbers of our tents?

A problem I’m working on. I think I have a solution, but I’m not sure.

• You have to define "reverse a number", and you have to clarify that all tents are numbered sequentially.
– smci
Jun 25 '20 at 5:30
• Plenty of answers depending on what "reverse" means and what numbers are allowed. Are decimals allowed? Tent 8.0 and tent 0.8, difference 7.2? By lowest numbers of our tents, do you mean which number is on the tents or how many tents there are? Jun 29 '20 at 12:28

Lateral thinking:

They have a central tent 0, and they numbered tents to the right - and tents to the left +.*

You and twin stay in tents

-1 and +1

• Kind of like streets in a city. You can live at 100 E. Spring street and 100 W. Spring Street.

• "1-" isn't really a number. Like the idea, though.
– Bass
Jun 24 '20 at 16:36
• @bass true, while $1 and 1$ are just regional display differences, still taken to mean the same thing, "tent minus 1" isn't how we count tangible things in reality so the answer should be constrained to positive integers. Jun 25 '20 at 1:37
• @Criggie sure we do not count items as “minus 1 tent” (except for in accounting I guess) but items can be numbered that way, again if information about direction was desired. Sure, they would probably call it 1N, 2N, 3N ans 1S, 2S, 3S. I did put lateral thinking and it was a joke answer anyway. Jun 25 '20 at 1:43

Using Roman numerals:

$$VI - IV = II$$

My solution: 91 campers. I assume “reverse” to mean the number “flipped” (so, the reverse of 10 is 01). 91’s reverse is 19, 91-19 = 72. Happens to be the smallest such numbers AFAIK.

• A trivial (?) pair is 08 and 80, which also has a difference of 72. Try some brute-force code online. I'd personally go with 8 as the lowest number and 80 tents minimum. Jun 24 '20 at 4:22
• Technically, the lowest possible number of tents could be just 2. Nobody said that the tents were all numbered sequentially. Of course the question was asking for the lowest numbers of our tents, not the total number of tents, so this is still correct. </wording nitpick> (Edit: Though I guess the title asks for how many scouts, so that could be just 2. Also, not every tent occupant is necessarily a scout.) Jun 24 '20 at 13:24
• I went with this one because I decided that 08 is not a number. Why not? Because that is what I decided ;-) Jun 24 '20 at 13:32
• @Hugh {Insert obligatory reference to using Base 3 here} Jun 24 '20 at 14:42

$$1$$ and $$3$$

Because:

You get the lowest possible numbers by numbering the tents base $$3$$. You're in tent $$01_3$$ and your twin is in tent $$10_3$$. Remember: always be prepared for a change of base! :D

• What about rot13(Ovanel)? The difference can be rot13(onfrgra). Jun 25 '20 at 0:24
• You beat me to it
– smci
Jun 25 '20 at 5:33
• @Damila That's far-stretched but it's not going to be any lower, in fact it's going to be rot13(orgjrra ryrira naq guvegrra). Jun 25 '20 at 10:21
• @Damila The only way to have an answer lower than this one would be for one of the tent numbers to be less than 1. (For example, if they were 0 and 2) Jun 25 '20 at 15:11

My solution: Considering two-digit numbers, their difference can be written as $$(10a + b) - (10b + a)$$, where $$a$$ and $$b$$ are positive integers representing the first and second digits of the larger number, respectively. Rewriting the expression gives $$9(a-b)$$. As their difference must end with 2, we can write it as $$10k + 2$$, where $$k$$ is another positive integer. The whole equation, then, is
$$9(a-b) = 10k + 2$$
Substituting $$n$$ for $$a-b$$ yields $$9n = 10k + 2$$, meaning we're looking for a positive multiple of 9 ending with 2. The smallest such is 72, and the rest take the form $$72+90m$$, where $$m$$ is yet another positive integer. Rewriting as an equation:
$$9n = 72 + 90m$$
$$9n = 8*9 + 90m$$
$$9n = 9(8+10m)$$
$$n = 8+10m$$
$$a-b = 8+10m$$
$$(10a+b)+(10b+a) = 8+10m + 12b + 10a$$
If by "lowest possible" the author meant their sum and you're not in the same tent as your twin, we can consider $$m$$ and $$b$$ to be 0*, giving:
$$(10a+0)+(10*0+a) = 8+10*0 + 12*0 + 10a$$
$$10a + a = 8 + 10a$$
$$a=8$$
The numbers, then, are 80 and 08
*Solving for b instead gives $$b=-8$$