Puzzles
Pocket money
When Dad gave out the pocket money, Amy received twice as much as her first brother, three times as much as the second, four times as much as the third and five times as much as the last brother. Peter complained that he had received 30p less than Tom.
Use this information to find all the possible amounts of money that Amy could have received.
Always a multiple?
Source: nrich
Take a two digit number. Reverse the digits and add the result to your original number. Your answer is multiple of 11.
Prove that the answer will be a multiple of 11 for any starting number.
Will this work with three digit numbers? Four digit numbers? \(n\) digit numbers?
Cycling digits
I have in mind a number which when you remove the units digit and place it at the front, gives the same result as multiplying the original number by 2. Am I telling the truth?
Mean, median, mode, range
A Find five one-digit positive integers which have a mean of 4, mode of 6, median of 4 and a range of 5.
B Find five one-digit positive integers which have a mean of 3, mode of 1, median of 1 and a range of 8.
C Find five one-digit positive integers which have a mean of 3, mode of 2, median of 2 and a range of 5.
Three digit numbers
Source: UKMT Pink Kangaroo 2012
Brigette wrote down a list of all 3-digit numbers. For each of the numbers on her list she found the product of the digits. She then added up all of these products. Which of the following is equal to her total?
A \(45\)
B \(45^2\)
C \(45^3\)
D \(2^{45}\)
E \(3^{45}\)
Multiple sums
Source: Project Euler
If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.
Find the sum of all the multiples of 3 or 5 below 1000.
Complex squares
For which complex numbers, \(z\), are \(\mathrm{Re}(z^2)\) and \(\mathrm{Im}(z^2)\) both positive?
![](javascript:showlimage("argand.png"))
Adding bases
Let \(a_b\) denote \(a\) in base \(b\).
Find bases \(A\), \(B\) and \(C\) less than 10 such that \(12_A+34_B=56_C\).