Todays puzzles are mostly written by Arsenii Nikolaiev, a Ukrainian mathematics undergraduate at Trinity College, Cambridge. Arsenii is a former silver medallist at the International Mathematical Olympiad, who, together with his brother Andrii, founded a maths and coding outreach club for Ukrainian schoolchildren, Kvanta, from which the problems below are taken.
Ukraine has a strong tradition in mathematics. In order to ensure that this will continue during and after the war, a group of Ukrainian mathematicians last month announced the creation of the International Centre for Mathematics in Ukraine, an institution that will support high level research and train young scientists. They believe that maths will have a key role to play in the reconstruction of the country once the conflict is over.
The location of the centre is yet to be decided, but the project will be paid for in part by Alex Gerko, the UK’s largest tax payer, whose algorithmic trading firm XTX Markets is the founding donor, pledging up to €1m in matched funding. The centre has the support of the European mathematical community, and six winners of the Fields Medal (the most prestigious prize in maths) attended the announcement, including the Ukrainian Maryna Viazovska, who won the medal last year.
One of the objectives of the centre is to support talented Ukrainian schoolchildren – through outreach initiatives like Kvanta, which produced the puzzles below.
1. Match sticks
Remove a single match from its position in the equation below, 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.)
Arsenii has given this puzzle to many friends at top universities and has found that the better you are at maths, the harder you find it to crack. Non Olympiad students, he says, do better than the Olympiad medallists.
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 ‘+’ and ‘x’ symbols, and you can use as many of either symbol 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.
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 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.
Note: these first three problems were all written by Arsenii
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?
Note: This problem was written by Arsenii and Andrii, whose mother is called Viktoria, and who have a sister. The actual ages of these real people almost matched the ages in the puzzle when the puzzle was created.
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).
Note: the authors of this problem are Sasha Tolesnikov, Vadim Koval and Fedir Yudin.
Ill be back at 5pm UK with the solution.
UPDATE: You can read the solutions here.
PLEASE NO SPOILERS
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.
