On Friday I set this puzzle…

What’s the fewest number of marks you could put on a 12-inch ruler and still be able to measure every distance from 1 inch to 12 inches?
If you haven’t tried to solve it, have a go now.  For everyone else the answer is after the break…
Put marks at 1 inches, 3 inches, 7 inches, and 11 inches.  Any other solutions?
I have produced an ebook containing 101 of the previous Friday Puzzles! It is called PUZZLED and is available for the Kindle (UK here and USA here) and on the iBookstore (UK here in the USA here). You can try 101 of the puzzles for free here.

1. All you need is a single mark at 1 inch and you can then measure any integer distance you like up to and beyond 12 inches.

1. Kaggy says:

Same here, only needed that 1 inch mark

2. JeffJo says:

No, you *calculated* a distance of N inches by subdividing it into N smaller distances that you measured individually.

3. @Jeffjo nowhere in the question did it say that you had to measure the distance using a single interval between marks on the ruler. It’s perfectly possible to measure (say) six inches using a one inch interval and sliding the ruler along five times using where the 1 inch marker falls as a reference point. Have you never used a ruler to measure beyond it’s nominal length?

Not my fault the question was not worded so as to exclude that possibility.

4. SagiK says:

I got this answer too. Just need to have 1 inch.

5. JeffJo says:

Steve, nowhere in the definition of “measure” does it say how to measure two distances, and combine them into one length. If you want to be this pedantic about the problem statement, you have to accept other pedantic comments that disprove your point.

6. Police Bureau says:

Er, if you’re not allowed to do more than one measurement, JeffJo, then how does the given answer work for 12 inches?

7. JeffJo says:

The same way it works for 1, 3, 5, 7, 9, and 11 – you can use the ends of the ruler as though they were marks.

8. Putidae Caseum Anthropophagos says:

JeffJo: Much as I’m enjoying your wit and erudition, I should point out that your post of 26 August at 9.39 p.m. is flawed in its use of ‘disprove’.

2. DiscoveredJoys says:

I came up with the same answer as Steve Jones, except I counted the ‘0’ mark as well, making 2 marks in total.

1. Anonymous says:

Correction: This problem is more aptly described by a sparse ruler. Notably, see the entry under length 12 for the complete solution set.

1,2,3,8
1,2,6,9
1,3,5,11
1,3,7,11
1,4,5,10
1,4,7,10
1,7,8,10

3. Anonymous says:

there isn’t really a break between the question and the solution!

1. Dave says:

There was a break of three days.

2. Mike Torr says:

Yes, this is something that always puzzles me. Every week it says “answer after the break”, but I don’t know what break that refers to. Am I just using an inadequate browser to read the page?

3. Smylers says:

The break only works for people who read the posts on the blog home page at https://richardwiseman.wordpress.com/blog-2/ where only the bit before ‘the break’ is displayed; to read the rest you have to click through, to this page.

1. Anonymous says:

This corresponds to the 1,4,7,10 sparse ruler.

1. Anonymous says:

This is identical to Richard’s solution, up to rotation.

4. Alex says:

Took me a little to work it out, mostly by trial and error, but I got the 1, 4, 7 and 10 solution.

5. I think the limit of measures we can take is missed. Otherwise the one 1 inch mark – mentioned at the beginning – works. And I still insist that if the ruler is one inch wide we can do without any marks.

6. @Sceptic_Tank says:

Similar 4-marks solution from me as well, BUT, this assumes that the phyisical length of the ruler is exactly 12″; i.e. a measuring stick rather than a conventional ruler with end ‘gaps’. A conventional (over-size) 12″ ruler would need to include the 0 & 12 as well. I looked up ‘Golumb ruler’, but there were no solutions for 12″, so I was unsure if my second answer “6 marks” was correct. Today I find the ‘anonymous’ post above, with the link to ‘sparse ruler’ (a new term to me) that vindicated my alternative solution. So, thank you Anonymous.

7. Anonymous says:

if it’s a ruler i would just the distances marked on it already…

8. I thought it might be 1,3,9 using the same theory as the 40lb weight problem 1,3,9,27 to measure nails…

To measure:
1 = 1
2 = 3-1
3 = 3
4 = 3+1
5 = 9-(3+1)
etc.

1. One Eyed Jack says:

If you’re going to do that, then all you need is a 1 inch ruler.

9. My answer was the same as Steve Jones’ above. The FEWEST marks is one inch. It may not be the most efficient solution, but it is the technically correct solution as the ruler itself is 12 inches. Any other solution violates the call of the question.

10. It took me about 5 minutes of trial and error on Friday before getting 1,3,5,11.

But the actual question was not about where to place the marks; it was about how many is the *fewest* necessary?

I then took a while to think about the fewest marks needed, and eventually came to a proof that three marks is not sufficient, thus: assume that the 0 point on the ruler is A, and the 12 inch point is E. The three marks you put on the ruler are B, C and D.

Therefore the only measurements that can be made are:
AB AC AD AE BC BD BE CD CE DE
in other words, only 10 possible measurements. To do 12 you need a 4th mark.

1. Dave says:

Good work, sir. How ironic that the “loony” gives the wisest answer here!

11. Miss Chili says:

The number of marks needed would be only three: 1, 3 and 7. If it is actually a full 12-inch piece of wood, then to measure 11 inches, you’d need only from that end to the 1.

1. Miss Chili says:

How one can measure 8″ is the way one can measure 13″ or 5″ or even 4″: adding. 😛

2. But if you can slide the ruler along and add, then surely you only need the one mark at 1″? So either sliding and adding is allowed, in which case your answer is wrong (because only one mark is needed), or it isn’t, in which case your answer is wrong (because you can’t measure 8″).

3. Miss Chili says:

12. Pete says:

Looking at the question laterally as so many of Richards require would give you the answer 0. You wouldn’t need to make any marks on a 12inch ruler as it already has the marks on it.

1. One Eyed Jack says:

Correct. If it isn’t already marked, then it’s not a ruler.

2. And that was my answer.

13. Ze Kraggash says:

I got the 1, 4, 5, 10 but Anonymous has that included in the list of 7 solutions…

14. I ended up with 2,4,5,11 by trial and error. Left out the “1” as it seemed superfluous with “11” in the mix. Still, couldn’t pare it down to three marks

15. Dougal McTavish says:

How do you measure the marks, unless you have a ruler?

16. Navneeth says:

What about 0 (3,6,10,11) 12 ?

1. Duncan says:

That’s the same as 1,2,6,9, just flipped round.

2. Navneeth says:

Right; thanks.

17. Duncan says:

Surely the answer is 4. Like John Cartwright I worked out 3 wasn’t enough, and 4 would be. I stopped there and didn’t work out the positions.

18. Anne Elk says:

I had a feeling that my answer of one mark would be perceived as a cheat, even it’s correct by the rubric of the question. I got the four marks answer as well.

19. I went for 1, 4, 8 and 10 but there are quite a few possibilities. My rudimentary maths suggests that you should be able to divide up a 15″ rule with just 4 marks as well, but I haven’t found a workable pattern yet. Any one any ideas?

1. Duncan says:

1, 3, 6, 10, 14 or 1, 4, 7, 10, 13 – I think that’s all of them (apart from flipping these around).

20. Bernardo says:

I think it’s better with 1, 2, 4 and 8 because you’ll only need as much, three numbers to calculate any distance.

21. Refuse to answer questions using barbaric units of measure; metric next time please.
Do you Anglos have 12 fingers? Why would anyone build a ruler base 12?

22. I got several “correct answers”!? … and it only gets more confusing when reading the comments in here.

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