Brain Teasers
13 cm Ruler
With a blank ruler (without marks) of exactly 13 cm you can only measure 13 cm. Now you are allowed to set 4 marks so that you can read off all full centimeter values from 1 to 13 cm directly from the ruler.
Where must these 4 marks be?
Much harder question:
The blank ruler has a length of 36 cm and you need 8 marks.
Set the marks.
Where must these 4 marks be?
Much harder question:
The blank ruler has a length of 36 cm and you need 8 marks.
Set the marks.
Answer
Marks on the 13 cm-ruler:1 - 2 - 6 and 10 cm.
You could then use these marks together to measure any length of 13 cm or less. For example, you could measure 8 cm by using the distance between the 2 and 10 cm marks.
Marks on the ruler with 36 cm:
1 - 3 - 6 - 13 - 20 - 27 - 31 - 35 cm
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Comments
The solution for the 13cm ruler is not unique. For example, 1-6-9-11 also works
Oops, no it doesn't!
cool teaser! I only got the first one.
I was right the first time. 1-6-9-11 does work.
By mirror imaging the two given answers, you also get 3,7,11,12 and 2,4,7,12
Btw, good teaser and good spotting Gizzer!
1, 4, 5, 11 also works (as well as it's reflection 2, 8, 9, 12). However, those are the only three (or six) ways. The 36cm ruler can only be done in the way given in the answer, or its reverse (1, 5, 9, 16, 23, 30, 33, 35).
For 3 marks on a ruler, there is a 10cm limit. With 0cm and 10cm and the 3 marks you get 5 different numbers (and 1 + 2 +3 + 4 = 10 different possibilities). For example: if you try 3 marks on a 12cm-ruler, you only get 10 different lengths (marks on 1, 4 and 10). Henry E. Dudeney thought, the limit for 8 marks is 33cm, and in 1926 he released 16 different solutions. That was correct 'til Leech 30 years later got the 3 more centimeters and with only 1 solution (without the mirrored). As far as I know, there is no formula to solve this problem in general. But, maybe I'm wrong. Nice day to all of you-Gerd
Btw: with the 3 marks on a 10cm-ruler not all lengths are possible (only theoretical).
I know a formula would be satisfying, but it's not really necessary when you can get a computer to check all the possibilities - that's how I found the 8marks/36cm solution
Solution shown is 1,2,6,10;
there are 5 others:
1,4,5,11
1,6,9,11
2,4,7,12
2,8,9,12
3,7,11,12
there are 5 others:
1,4,5,11
1,6,9,11
2,4,7,12
2,8,9,12
3,7,11,12
I COULD have written a computer program to solve this, but that's not much of a teaser that way.
After a lot of effort I found the 36 cm rule (actually the mirror of the one given). I found both 1,2,6,10 and 1,4,5,11 for the 13-cm ruler.
Fun and challenging teaser!
After a lot of effort I found the 36 cm rule (actually the mirror of the one given). I found both 1,2,6,10 and 1,4,5,11 for the 13-cm ruler.
Fun and challenging teaser!
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