Make 60 by Using the Same Number Thrice
Here is another riddle I discovered in a book and gave as homework to my students.
Puzzle. I can use the number 20 thrice to make 60: 20 + 20 + 20 = 60. Make it 60 again by using a different number three times.
The book’s answer was to use 5: 55 + 5 = 60.
My students were very inventive. All of them solved the puzzle, but only one out of ten students came up with the book’s answer.
- For most of them, the new number was 60, as in 60 + 60−60 = 60, or 60*60/60.
- Some used −60 or 1/60, as in −60 − (−60) − (−60) = 60, or ((1/60)/(1/60))/(1/60) = 60. Similarly, some multiplied the cube root of 60 three times.
- One student used 59 in a clever way, as in 59 + 59/59 = 60.
- Another student said the following. If you turn 60 upside-down, you will make 09, and now you can use the number 3 thrice: 3+3+3 = 09.
- And the last on my list is the student who said that 42 has to be the answer to the universe and everything. He summed up two instances of 42 to get 84 and then subtracted the third instance of 42 with the digits flipped to get 84 − 24 = 60.
Ivan:
60 = 66-6 = 60^(Log(60,60))
Paul Dirac’s formula is the best
5 November 2024, 1:14 pm60 = -Log(2, Log(2, Sqrt(Sqrt(Sqrt(…(2))))))
60 times
Leo B.:
If we’re talking about numbers rather than digits, the notation should not matter. We can say “twenty plus twenty plus twenty is sixty”, for example, or “fifty-nine, plus fifty-nine divided by fifty-nine is sixty”. That said, the book’s solution is incorrect. There is no way to spell out the proposed answer while using just the number “five” three times.
5 November 2024, 3:53 pmtanyakh:
Leo, we can look at 55 as a concatenation of two numbers 5 and 5.
5 November 2024, 3:57 pmLeo B.:
How are you proposing to spell that out, to make the expression based on notation-independent numbers? “Five concatenate-decimally five plus five”? That has the number 10 in it.
5 November 2024, 6:57 pmLazar Ilic:
In a relatively popular variant of 24 played with Patrick Guo at Texas ARML American Regions Mathematics League, exponentiation and factorials are allowed, and the target number can change. Here, (3!)!/(3!+3!) and 6!/(6+6) work. As in your student’s 59 solution, one obtains 61-61/61.
7 November 2024, 9:48 pmAndreas:
This may be bending the rules a little: (.9… + 9) * 6
8 November 2024, 12:02 pmDavid Reynolds:
[(sqrt(241)-1)/2][(sqrt(241)-1)/2]+[(sqrt(241)-1)/2]
9 November 2024, 1:15 amDavid Reynolds:
Here’s a totally out of the box solution.
9 November 2024, 12:20 pmIn Roman numerals, take V, V, V.
Turn the 1st V on its side to make an L.
Turn the second V upside down and place the third on top to make an X.
LX in Roman numerals is 60.
tanyakh:
Wow! Thank you, everyone, for so many interesting solutions. I especially like out of the box ones.
Here is a straightforward solution from my son, Alexey: sqrt(30)sqrt(30) + 30.
9 November 2024, 12:26 pmIvan:
60 = 60*Cos[60^60 Degree] = 60*Cos[60*60 Degree]
9 November 2024, 3:59 pm