# Back at the PSE headquarters

Roaming through the PSE building I come across a solid steel door. Above the door is a sign saying “Astrodome R”. I had no idea this place delved into astronomy.

Below this sign is a painting of sorts:

Obviously abstract, but I kind of like it. Below the painting is a little rhyme:

Roses are red
Violets are blue
Least pushes is best
Binary is good too

Hmm.

At the side of the door is a four digit display currently showing 0000. Below the display are two push-buttons. The left button has imprinted the text "x10" while the right button has the text "+7". Above the display is the text "Immediate calculation. Only 4 LSD shown".

LSD? Is this perhaps a storeroom for drugs?

Can you help me figure out how the 2 buttons should be pushed to open the door? And what might be behind the door?

Hint:

Hmm. Four rainbow colored sections in the painting and four digits in the display. Could there be a connection?

Hint 2:

I take a chance and try the door handle. Locked, as expected. In frustration I randomly press the two buttons for a while. The display shows 1249. The door remains locked.

• Does the display go back to 0000 at any time after the button presses have begun?
– mjjf
Sep 16, 2020 at 6:02
• @mjjf No, it does not.
– Jens
Sep 16, 2020 at 18:32
• May be a silly question, but if I try to open the door without touching the buttons at all, what happens?
– mjjf
Sep 17, 2020 at 4:45

Assuming @mjjf in on the right track Re: LSD,

We need to find the shortest route ("least pushes is good") to a number ending in 1248.
Since 7 and 10 are coprime, the last digit of our number is uniquely determined by the length of our last run of +7's. Also, we can never save presses by putting off a ×10 for later, because to get our last digit correct again we would end up doing an additional 10 +7's and an earlier ×10 can do that cheaper.
To get an 8 we need 28 which requires us to have X122 already, to get a 2 we need 42 which requires X08, another 8 requires 28 and X8, and likewise once more, so our code looks like
+7×4 ×10 +7×4 ×10 +7×6 ×10 +7×4
which results in (3)1248 as desired.

• Congratulations! The door is now unlocked. :) As you got the final answer, I'm giving you the checkmark, but please upvote @mjjf's answer as he did most of the work.
– Jens
Sep 18, 2020 at 16:29

It's a Takuzu. Solution below:

After that:

Once solved, the tetronimoes are 4 digit binary codes, probably written from left to right following the shape. The hint in the question helps to know what to do with them: "Four rainbow colored sections in the painting and four digits in the display. Could there be a connection?". Here's what I got for each:

red: 0001 yellow: 0100 orange: 0010 green: 1000

re-arrange to match the rainbow color order (ROYGBIV): red, orange, yellow, green --> 0001, 0010, 0100, 1000 (infinite series of powers of two)

Then:

The display is basically a calculator with only two buttons. One button multiplies by 10, the other adds 7. The hint seems to explain that each tetronimo is a digit in a larger number. Our number is 1248. Now the part that isn't clear - will the display only show 1s and 0s, or base 10 numbers? Note that I assume LSD is referring to Least Significant Digits, meaning the display only shows the least significant digits.

If binary: Well, 1248 in base 2 is 10011100000. And the last four digits are 0000...which is the current display. So, maybe the door is already open and we just need to turn the knob?

If decimal: We need the display to say 1248 and we have to get there with only these two buttons in the least amount of presses. I'm not sure if this is possible with just adding and multiplying by 10 since 1248 and 7 don't share any common factors besides 1. I am not a mathematician, and I don't have an answer for this yet. I know that multiplying 0 by 10 is pointless, so the first step must be adding 7. But I will update if I find a suitable answer.

And what might be behind the door?

I guess this has to do with the context clues. As was pointed out by Deusovi, Astrodome R is an anagram of "moderator" and PSE is Puzzling Stack Exchange (I feel silly 🤦‍♀️). So I guess we would find the moderators of PSE!

• PSE is this site; Astrodome R is an anagram of "moderators". One other possibility for the calculator: take red, orange, yellow, green together as binary and convert to decimal/hex, and then make that whole number using the calculator (which works in decimal/hex).
– Deusovi
Sep 16, 2020 at 6:10
• Just checked; the solution isn't unique as a binary puzzle, which is why you may have been having trouble.
– Deusovi
Sep 16, 2020 at 14:37
• @Jens Ah, never mind! It does look unique - I missed two givens.
– Deusovi
Sep 16, 2020 at 18:47
• Nice work mjjf. Could you please upload the solution to the takuzu? Sep 17, 2020 at 9:31
• Good progress! I added a hint and a math tag. Maybe the math tag will attract one of the site's math wizzes to help out?
– Jens
Sep 17, 2020 at 20:50

I think @mjjf mostly has it figured out. The issue then is

The order of the target number. Although red-orange-yellow-green spells out 1248 as the other answer found, perhaps we don’t actually need to rearrange to ROYGBIV order.

Specifically,

The colours are in the order Red-Yellow-Orange-Green. In this order, the target number is actually 1428. Note that this number actually appears to be reachable using x10 and +7 with only a few button clicks, namely: +7 +7 x10 x10 +7 +7 +7 +7. I’m thinking that 8 button presses is minimum to achieve this.

Now,

Does the door to the moderators open after pressing this sequence??

• The door, obstinately, remains locked. ;)
– Jens
Sep 17, 2020 at 21:23