Earlier today I set you these problems from Kvanta, a Ukrainian maths and coding club founded by the brothers Arsenii and Andrii Nikolaiev. Here they are again, this time with solutions.
1. Match sticks
Remove a single match from its position in the equation above, and place it in another position so that the equation is correct. (This is not a trick question, so moving a match to mean ‘≠’ is not allowed.)
Solution – 2 + 5 = 3
2. What did seven do?
Eight nines, of course! And so here are eight 9s in a row
9 9 9 9 9 9 9 9
Place the arithmetical symbols ‘+’ and ‘x’ between some of these 9s in such a way that the total value of the expression is 9999.
You don’t need to use both symbols, and you can use as many of the symbols as you like. If there is no symbol between two digits, adjacent digits combine: thus ‘9+9 9 9’ would be ‘9 + 999’. You can use brackets to be clear which operations must be done in which order.
Solution There are a few, such as
(999 x 9) + 999+9 or (99 x 99) + 99 + 99
3. Robot swap
The 3x3 grid above has three walls around the middle cell like a ‘c’. Robots X and Y are placed on the cells indicated. Come up with a program that is made up from only these four commands,
move one cell up
move one cell down
move one cell right
move one cell left
such then when you run the program simultaneously on both the robots, the robots change places. (Or prove it isn’t possible.) If a robot cannot obey a command because there is a wall in the way, or it is the external border of the grid, the robot skips that command.
Solution up, right, right, down, left, left, up.
4. Family relations
Viktoria’s age now is equal to the combined ages of her daughter and two sons. In several years time, Viktoria’s age will be equal to the combined ages of the two sons. How much older will the daughter be at that point in the future than she is now?
Solution Twice.
Let the ages now of Victoria, the daughter and the sons be V, D and SS, and let the point identified in the future be x years away. We know that V = D + SS, and that V + x = SS + 2x.
This second equation reduces to V = SS + x. Thus x = D, and the daughter will be 2D in the future, or double her current age.
5. Mystery symbols
Senya knows that the symbols Ⰰ, ב, and ௩ mean 1, 2 and 3 in some order, but he is not sure which symbols means what exactly.
Help Senya make an expression using these symbols, and the arithmetic operations +, x , and – , so that the value of the expression is definitely equal to 1. No division is allowed. (For example, if you write ‘Ⰰ x ב’ then this expression can be equal to 2, 3 or 6)
Solution
It was cool to use curious characters, but let’s call them a, b and c for simplicity.
We know that a + b + c = 1 + 2 + 3 (in some order) = 6
And we know that a x b, a x c and b x c are, in some order, 2, 3 and 6. Thus
(a x b) + (a x c) + (b x c) = 2 + 3 + 6 = 11
Thus a + b + c + a + b + c – ((a x b) + (a x c) + (b x c)) = 1
I hope you enjoyed today’s puzzles, I’ll be back in two weeks.
Thanks to Arsenii Nikolaiev. To find out more about his Ukrainian outreach maths club Kvanta, click here.
I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
I give school talks about maths and puzzles (online and in person). If your school is interested please get in touch.
• This article was amended on 8 February 2023. A previous version misnamed Andrii Nikolaiev as Andrew. This has been corrected.
