From Wikipedia:
Runways are given a number between 01 and 36. This indicates the runway’s heading: A runway with the number 36 points to the north (360°), runway 09 points east (90°), runway 18 is south (180°), and runway 27 points west (270°). Thus, the runway number is one tenth of the runway centerline’s magnetic azimuth, measured clockwise from the magnetic declination.
I was recently flying over Omar N. Bradley airport in Moberly, MO and noted that its runway was numbered 31/13. I assumed that there were a runways where each direction would be a mirror of the digits of the other direction, but a quick glance around the compass proved me wrong. I then hypothesized that 310 and 130 were the only such directions. My dad quickly found that 200 and 20 (displayed as 20 and 02) was another. This is what weird people do. But particularly weird people make a spreadsheet.
Yep, those are the only two. Column A is the numbers 1-36 formatted as “00?. Column B has the formula
The conditional formatting for A2:B37 has this formula
Is there a better way to test that?
Sounds like a solver problem, actually. 10a + b = 10b + a + 18 mod 36; a, b =>0, a, b < 10.
One way:
B2: =A2+18-36*(A2>18)
CF: =RIGHT($B1,1)&LEFT(TEXT($B2,”00?),1)=TEXT($A2,”00?)
Oops – should have been $B2 in the first CF reference
Just created a 360-row spreadsheet to see whether there were any such instances in the three character versions. I.e. 12 degrees = 012 that has an opposite direction of 192, reverse is 291. There aren’t any.
We can solve this problem mathematically. We have a two-digit number. Let L be the larger digit and S be the smaller digit. Then one of the relationships is this…
10L + S = 10S + L + 18
Subtracting L from both sides yields this…
9L + S = 10S + 18
Subtracting S from both sides yields this…
9L = 9S + 18
Dividing both sides by 9 yields this…
L = S + 2
This tells us that the larger digit must always be two more than the smaller digit. Okay, for later use, let solve this equation for the smaller digit…
S = L – 2
Now, the next relationship is that the larger number cannot be greater than 36 (since there are only 360 degrees in a full circle). So, the expression for this is…
10L + S < 36
Now, let’s substitute the equation for S that we found earlier into this relationship…
10L + L – 2 < 36
Performing the indicated math yields this…
11L – 2 < 36
Add 2 to both sides yields this…
11L < 38
Divide both sides by 9 yields this…
L < 3.4545….
Since both digits are whole numbers that are not negative, then there are only two values for the larger digit that are less than 3.45+ and for which the smaller digit is 2 less than it (remember, our first equation told us the larger digit was always 2 more than the smaller digit)… those numbers are 2 and 3. So, the only possible larger numbers are 20 and 31… no other possibilities meet the given conditions.
So I ran solver (needed to add a constraint that 10a+b and 10b+a were <=36) and came up with 2,0 as the possible results. Excluded a and b from being 0 and came up with 1,3. No other results.
Sounds like a great puzzler for CarTalk.
While it’s possible to solve this algebraically, it’s odd that anyone familiar with accounting would be unfamiliar with the fact that two numbers with adjacent transposed digits differ from each other by multiples of 9, e.g., 1234 and 1324 differ by 90, 1729 and 7129 differ by 5400.
2-digit permutations which differ by 18 are those with separate digits differring by 2: 0-2, 1-3, 2-4, 3-5, 4-6, 5-7, 6-8, 7-9. Only 0-2 and 1-3 produce both 2-digit numbers in [1,36].
@ Dick
I see I had a typo in my posted mathematical solution to this problem. Towards the end of my message, I had these three lines…
11L < 38
Divide both sides by 9 yields this…
L < 3.4545….
If you were wondering why I said to divide both sides by 9, then you have company because I have no idea why I said that either. The obvious division is by 11, not 9… the indicated originally posted resulting inequality was correct though… 38 divided by 11 is indeed 3.4545…
Dick, this is your seventh post in two weeks (and some are pretty substantial) – I haven’t been able to keep up with reading them! Are you either out of work or suffering from insomnia? :-)
/Roy
[…] Daily Dose of Excel » Blog Archive » Runway Math […]
@Roy “Are you either out of work or suffering from insomnia?”
Neither. He is keeping me amused/informed! (grin!)
I like to lull you to sleep with sparse posts. Then, when your expectations are reasonably low, I come out with some crap. In reality, I go through streaks. Sometimes I have a ton to say and sometimes nothing. I wish I knew what triggers that.
Two comments.
Because of symmetry, the numbers in the 1st column only need to go up to 18. Hence, the formula in the 2nd column becomes =I2+18 (my first column of data was I starting with row 2).
Unlike most, I prefer functionally equivalent arithmetic operations over string operations. I don’t know the comparative costs (and it definitely wouldn’t matter for such a small sample), but one reversed digits test is =MOD($J2,10)*10+INT($J2/10)=$I2