Legal 21, Officer
Young Saul, a budding mathematician and printer, is making himself a fake ID. He needs it to say he’s 21. The problem is he’s not using a computer, but rather he has some symbols he’s bought from the store, and that’s it. He has one 1, one 5, one 6, one 7, and an unlimited supply of + – * / (the operations addition, subtraction, multiplication and division). Using each number exactly once (but you can use any number of +, any number of -, …) how can he get 21 from 1, 5, 6, 7?
Note: you can’t do things like 15+6 = 21. You have to use the four operations as ‘binary’ operations: ((1+5)*6)+7.
Problem submitted by [email protected], phrasing by yours truly.
Email me at sjm1 AT williams.edu for hints
Correct. Can use to group, cannot use as binomial (though a clever idea!)
I see that you use parentheses in your explanation of the problem. But the wording leads me to assume that the binary operation of “n choose k” is not one of those allowed. Correct? But parentheses, as many as needed, are allowed? Thanks.
I can’t figure it out and I want to give it to my students. can you please email me the answer!!!
Thanks,
All emails to you bounced — please email me at [email protected]
5!/(7+1) +6. I know you can’t use factorial but cmon, this is awesome
i realy enjoyed read all article and posting in this blog , and i like this website design with light purple backgroud ,
nay but very clever!
1+5+6+7 = 21 in base 9. Yea or nay?
very original, but you can solve without introducing a decimal point.
Not sure, but:
(5 x 6 x 7) x .1 = 210 x .1 = 21
Does it involve turning a number over (like 5=>2)
Then it could be 2*7+6+1
email me: sjm1 AT williams.edu
sure: email me: sjm1 AT williams.edu
hint please 🙂 Spent too much time on it..
Hello I know the answer to this problem but I want it to give to my students but what kind of hint can I give them so they can help them to come faster with the answer. Thanks.
no — there is a way using +-*/
highly original, beautiful answer, but there is one using +-*/
hello!!!very nice riddle steve!!!i was thinking about the solution and something came up but i dont know if its legal with the rules of the riddle..my thought was: “7 choose 5” + log6(1) where “7 choose 5” is the combinations of seven to 5 and log6(1) is the logarithm to the base of 6
Could u do 1*5*6*7=210 and just drop the zero?
yoav: well done — email me at sjm1 AT williams.edu for more on this problem
sure
Can i have a hint for this one, I’ve been trying For a few days now and can’t get it. I’m thinking it has to do with fractions am I on the right track?
sorry — I don’t want to post the hints — can you get an email account? sjm1 AT williams.edu
i don’t havde an actual email so could u just post it??
let me send a hint first. /s
I can’t solve this after thinking about it for hours! Can I have the solution please?
sure, I sent a hint (you can also email me at sjm1 AT williams.edu)
I am pulled this problem for my class and no one figured it out. Could you please send me the solution. It would be greatly appreciated.
sure, sent
I need a hint, I’m trying to give this riddle to my dad but I cannot figure out the answer my self. He gave me a difficult riddle about 15 minutes ago and I couldnt figure it out. So this riddle is my Revenge! so help please. 🙂
will send a hint
I’ve spent too much time thinking about this, please send me the solution. Thanks.
sure, sending //s
May I have a hint?
sure — sent
pls give me a hint
sure. I’m working on adding a student/teacher corner, and wondering if you and/or your students are interested in helping. Details follow:
I have several goals for the web site. Primary, of course, is that riddles are fun, and a great way to keep you mentally active.
Another hope, though, is that these will be of use in our schools. Many of these riddles are appropriate for high school and
junior high school settings. These aren’t just for the best and brightest (though these can be something for them to look at
while teachers have to review other material), but a chance to help everyone think more creatively. We get so bogged down in
passing exams and being taught to exams that we lose the sense of exploration. We learn or are taught the mechanics; this helps
us become technicians, but more is needed. We want to learn how to ask questions, to see where things can go….
A lot of these riddles are related to great concepts in modern mathematics, though the connections are not always apparent.
That’s one of the goals of the student / teacher corner, to talk about these connections and try to excite people. Math is not
an old, dusty, dead subject — it’s alive, with issues and wonrous applications. For example, the ‘C is for Cookie’ problem is
related to some of the hardest partitioning problems in number theory, while the chess problem is related to designed a baseball
schedule or an airline schdeule. A big part of the student / teacher corner is to make these connections explicit, and provide a
guide to further reading.
Another goal is to describe the thought process behind attacking and solving these riddles. What is a good thing to try, and
why. This includes not only giving the right answers, but also the wrong ones. What was good about that wrong answer?
Anyway, I hope this gives some sense of what I hope we can acocmplish.
May i have the solution? I am planning on giving this problem to my high school for a “math month” riddle
sorry, no square-rooting. you can do it without.
Is it (7*6)/square root of 5-1? or no square roots allowed either ?
beautiful — great thinking outside the box, but there is a way to solve it using the 6 as a 6.
If we can flip the 6 to make it a 9 then (9-5-1)*7 or 19-5+7
I’m missing something: isn’t this just 30 – 7 – 1 = 22, not 21?
5*6-7-1
I like it — great ingenuity. sadly, we don’t have a decimal point….
1 x 6 x 7 x .5
there is a standard soln involving just +-*/ as the operations are intended.
he has an infinite number of / and – right so if you put those two shapes together (using >1 “-” sign if nessary” you can make a 7 shap repete the proscess and you get 7+7+7. and before you say you can’t do that… the question used symbols that arn’t allowed… like “(” and “)”
Nice originality, but you can do it without splitting up something like 13 — you’re supposed to just use the four numbers and the standard operations.
i know the answer now. 7+6=13 i separates the 13. i got 1 and 3 right?
i 1+3=4*5=20+1=21!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
very nice try, but you can’t do this as you don’t have a 0
how about 70/5=14+6+1=21
i can do that right
Nice try, but it can be done without a decimal (as that is introducing a new symbol).
First instinct: 6 * 7 * 1 * .5 (= 42 * .5 = 21) Decimals are not permitted?
Kher ([email protected]): hadn’t thought of that soln, but nice!
correct kareem — not posting as it’s the soln
you can do it without that; however, if you can do it that way, I’d love to see (email me at [email protected])
can we arrange to numbers to make a bigger number? for example, using 1 and 5 to make 15?
No, but a very original attempt!
is the answer 2.1 (1+5)/(6+7) 6/13 2.1666666
Not quite — the numbers are 1,5,6,7 not 1,3,6,7
Done. (6/3+1)*7
Going to, or is? Email me at [email protected] for a hint.
This is going to drive me crazy.
no, only +-*/
Can we use exponents?
doesn’t give exactly 21 with all 4 numbers being used, and is thus not the soln
Could he just say he’s (7+5)/6 1? I’m not completely sure I understand the question. The value of (7+5)/6 is 2, so by stating you’re (7+5)/6 1, this could be read as 2 1, or twenty one.
Aldo: correct
If this is 1/(5*6*7) or (1/5)*6*7, neither gets you to 21. look at all the different ways you can structurally combine 4 numbers….
1/5*6*7? Still thinking on the 1,2,3 Q, hard.
To: Yair.K: correct solution, glad you’re enjoying the riddles. I’m not posting your reply as it gives the answer.