Friday quiz maths question 26 Aug 2022

Today's question is 

I’ve thought of five numbers.  If I three of them together in each possible combination, the totals I get are 10, 14, 15, 16, 17, 17, 18, 21, 22 and 24.  What are the five numbers?

I'd love to see an answer involving algebra. I started with the assumption (correct, I believe) that the lowest three of the five numbers will add to 10, and the highest three will make the 24. 

So we've got A+B+C = 10 and C+D+E = 24 (if A-E go lowest to highest). We can get something from this (D+E)-(A+B)=14 and 2C+A+B+D+E = 34

It's interesting that two combinations make 17 but that's all I've got in that direction.

[Future me: I did a bit more work on the algebra and thought I was getting somewhere when I boiled those things down to 2C+2A+2B=20, then realised that's the A+B+C=10 that we started with! D'oh!]

So I did my usual thing and wrote a program to "try all of the numbers".

As I worked I realised that we don't know whether all of the numbers are different, whether they're integers, whether they're positive, whether zero might be one or more of the numbers (I would argue that zero isn't a number).

So as a starting point I made those assumptions - positive integers, all different. And I got an answer that seems to fit.

Here's my program, I happened to have my sixtyclone out and hooked up. It's pretty longwinded but hopefully reads OK because of that. The first few lines set up the loops, and they assume that each number is at least one more than the last. 


It tests for exactly two combinations that make 17, and  the lowest three making 10 and the highest three making 24.

In a matter of seconds it gives three possible answers. It was quick work to find which one of those can give 14, 15, 16 etc  with a combination of three. (It's the last one:  2,3,5,9,10)

The official answer comes in later but those seem to fit. 

Do you have a neater way to do this? Yes I could make my basic program more elegant, but is there a quicker way to find the answer, particularly in algebra?



Comments

Post a Comment

Popular posts from this blog

e-ther 2 - Announcement and full rundown of features

Image
I'd like to announce the new and improved e-ther 2. This information is available in video form here . If you're not familiar with e-ther, then it's a small theremin-like instrument with MIDI and audio out . At first glance the differences aren't obvious and indeed the big changes to the hardware are internal.  However, I hope you'll find the list of new features as exciting as I do.  Hardware-wise, I've put the microcontroller and DAC onto the main board.  Although this increases costs (for me, not for you), there are some big advantages. Besides looking more professional, the audio module is no longer a 'buy separately' optional extra. It's incorporated and the audio jack is just there if you want it.  The audio module used to fit below the board but with the audio capability now built-in, the profile is more compact. Most of the new features are down to firmware which means that a large proportion of what I'm calling 'e-ther 2' will be
Read more

Enclosures for MIDISID

Image
 The default configuration for MIDISID  looks fine and is functional. I have fitted wooden cheeks to my own, which gives it a look and tilts the top forward, making it a little easier to use when you're sitting back from it. MIDISID was designed from the start with 'bring your own box' in mind. It's a modular design, the top and bottom parts are easily separated and connect with a 4-way jst-sh cable. In my head I saw it being used in a sloping-top case. Paul of Bad Dog Designs has come up with a different solution. It achieves the same goal - tipping the controls and screen forward. It's a 3D print in two parts. It takes zero effort to fit - MIDISID just drops in. I don't know whether he has made the design files public but  please contact him  if you're interested. Update Nov '23 The current version of MIDISID is now larger, has a tilted front and custom side supports: Owners are still being creative with enclosures. Here's a beautiful one from  @k
Read more

MIDISID - second batch and giveaway

Image
The first MIDISID batch ('earlybird' pricing) have now all gone, I have the materials for the second batch and the first of those are now made.  There are now black and purple editions. At risk of going all Tim Cook, the black really does look gorgeous. Hardware-wise they're hardly altered but the firmware ('MIDISID OS') has come a long way, new features include: you can set glide speed modwheel support; choose vibrato, filter cutoff or pulse width Increases the range of vibrato, to enable spooky effects Now supports RPN 0 messages - for setting the pitch bend range. (This was important for getting my tiny MIDI theremin working - see https://twitter.com/midi_in_out/status/1574870874158764035 ) ( The manual is here in case you'd like to read about all of the features.) I'll shortly be mailing those who have bought MIDISID already with details for the new firmware. Giveaway Because I'm an idiot, I (genuinely) accidentally bought a second copy of @C64Audio
Read more