Earlier today I set you up with three puzzles for 12 year olds, which are used by the charity Axiom Mathematicswhose mission is to help top-achieving, low-income children continue to perform well throughout high school.
1. Backward multiplication
Which four-digit number reverses itself when multiplied by 4? As in, what are the numbers a, b, c and d such that the number abcd x 4 = dcba?
(In this problem, the letters a, b, c and d all stand for different numbers.)
Solution 2178 x 4 = 8712
STEP 1 a can only be 1 or 2, because four times a number greater than 3000 will be greater than 12,000, so has five digits, not four;
STEP 2 The solution must be even (since all multiples of 4 are even) and so on a should be 2.
STEP 3 We know that 4 xd is a number that ends in 2. Working through the four times table we arrive at d = 3 or 8
STEP 4 d can only be 8 or 9, since 4 x 2 thousand and something is at least 8 thousand and something; so d is 8; so 4x b (possibly plus a carry) is less than 10, otherwise d should be 9, so b = 1 or 2, and must therefore be 1 since it must be different from a.
STEP 5. c x 4 + carry of 3 has one digit 1, so c x 4 has units digit 8, so c can only be 2 or 7 and must therefore be 7 since it must be different from a.
2. Really secret Santa
A group of nine secret agents: 001, 002, 003, 004, 005, 006, 007, 008 and 009 organized a Secret Santa. The instructions are coded to keep the donors secret.
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Agent 001 gives a gift to the agent who gives a gift to agent 002
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Agent 002 gives a gift to the agent who gives a gift to agent 003
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Agent 003 gives a gift to the agent who gives a gift to agent 004
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and so on, until
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Agent 009 gives a gift to the agent who gives a gift to agent 001
Which agent will Agent 007 get her gift from?
Solution 002
The easiest way to do this is to draw a circle and then fill it in.
3. Trapezium, or stair difficulty?
Here is a trapezoid, with two parallel horizontal sides. Where would you place a vertical line to divide the shape into two parts of equal area?
Solution
Find the midpoint of each non-parallel edge. Find the midpoint of the line joining these two points. Place a vertical line through this point.
I hope you enjoyed these puzzles. I’ll be back in two weeks.
For more information on Axiom Maths, please read the original post or go to their website.
If you are a parent or a school interested in getting involved for September 2024, you can fill out the Axiom Maths form here.
I’ve been doing a puzzle here on alternate Mondays since 2015. I’m always on the lookout for great puzzles. If you want to suggest one, email me.