Children in in Changzhou, China, line up to form characters ‘2019’ during a performance to welcome the new year. Photograph: VCG/VCG via Getty Images (and The Guardian)
The UK newspaper, The Guardian, hosts a Monday puzzle column by Alex Bellos, and we have been trying out his first set of puzzles for the new year all based on 2019 & its digits. We’ll share some of our solutions, mathematical musings, and some of the questions that these puzzles brought to mind in the comments. Please feel free to add your own.
1) Date jam
(i) Using each of the digits 2, 0, 1 and 9, exactly once, create expressions that equal all of the numbers from 0 to 12. The expressions can include any of the arithmetical symbols +, –, x, ÷ and √, and brackets. [For example, 0 x (2 + 1 + 9) = 0]
(ii) Create expressions using 2, 0, 1 and 9 for all the numbers from 13 to 20. Again each number must be used once and only once in each expression, but as well as the operators mentioned above you can now use concatenation to make two-digit numbers (e.g. 2 and a 0 can concatenate to 20) and exponentiation (meaning that one number can be the power of another, such as 19)
2) Countdown conundrum
Fill the blanks in the following equation, so that it makes arithmetical sense:
10 9 8 7 6 5 4 3 2 1 = 2019
You are allowed to use any of the basic mathematical operations, +, –, x, ÷, and as many brackets as you like. An answer might look something like (10 – 9 + 8) x (7 – 6 – 5)÷(4 + 3 + 2 + 1) = 2019, but not this one since the equation is incorrect.
this a very interesting problem about 2019 and how you can only use two zero and nine and one to and the other problem was a little bit more easier because you could use the numbers between zero and ten
I’m glad you found that the restrictions in the problem made it challenging & interesting. What solutions did you find?
this was relly hard i cant figure out 13
What solutions did you find for 0 to 12? Did you find any for 14 to 20?
this worksheet was very hard
Sharing the questions you have or even describing what you found most difficult would be a good way to practise expressing & clarifying your ideas, and it might even lead to some help!
this was hard at the beginning because I didn’t know what to do but when it got explained it made it easy but i still have question for it. I noticed that the 0 can be added anywhere because 0+1=0, 0+3=0, 0-1=1, so the 0 is useless in this problem. It was hard for me to add the 9 most of the time because it was a big number but overall this was fun but challenging. 😛 🙂
I wonder whether you might be mixing up the symbols you are typing for different operations. Are you suggesting that zero plus one is zero?
It can be tricky to type out expressions & equations we are used to writing by hand, especially when you have a limited set of characters to use as in this commenting feature. On computers, we have familiar characters for addition and subtraction (+ & -), but we’ll often use different characters for multiplication & division. For multiplication, we can use an asterisk (*), and for division, we can use a forward slash ( / ) just like a fraction bar (which is a division symbol. In fact, you can use these symbols (+,-,*,/) as operators to perform calculations in programs like spreadsheets.
I will disagree with you about the usefulness of the zero in this problem. You are right that a large number might be difficult to use, but maybe that zero can help you get a big number out of the way. Try multiplication.
This problem seems really complicated and interesting. One question I have is what exponentiation is. I hope I can figure out the countdown conundrum question. It seems the hardest out of all of them!
Mr Bellos did use some vocabulary that is probably new to many of us, but there were also some examples to help us out. Did you notice the example given for exponentiation? There was a small 9 in superscript (raised up a little higher & placed just after the numeral 1). That example was 1 to the power of 9. We can write the same thing this way: 1^9. That example means multiplying 1 by itself nine times. So 1^9 = 1x1x1x1x1x1x1x1x1. And of course 1x1x1x1x1x1x1x1x1 = 1. You may remember us talking in class before about squaring numbers.
I’m glad you are finding the puzzle both complicated & interesting at the same time! Please share some of your solutions.
date jam: 0x(9-2-1)=0
9×1+(0+2)=11
Countdown conundrum: 2014+5=2019
10^3×2+19
Both of your equations work for the date jam problem. Can you find the rest?
I am delighted you were able to show exponents using a keyboard and this very simple text editor. You are right that 10^3 * 2 + 19 = 2019. Hurrah! Notice, however, that for the countdown problem, your equation has to use ALL the numbers from ten to one in descending order in your equation (like in the familiar New Year’s Eve countdown to midnight).
So far, so good; keep it up!
solving the problem was really fun and hard 😛
Please tell us more! What did you notice? What made it fun? Why do you think you were able to get pleasure out of a tough situation (one where others might experience frustration rather than fun)?
I think that 2019 is a great year for the future and I like that they where many people lining up by numbers. It is very cool and Thank you. and it is like a maths problem as well and it looks like that it is in china
The photo that appeared in the original article in The Guardian (which is the image I used here as well) is indeed from China. I am glad that you found the post, photo & problem interesting. Do you have any questions or suggestions that might help us find some expressions here?