Puzzles
Reverse bases again
Find three digits \(a\), \(b\) and \(c\) such that \(abc\) in base 10 is equal to \(cba\) in base 9?
Two
Find \(a\) such that \(a+(a+A)^{-1}=2\), where \(A=(a+A)^{-1}\).
ie. \(a + \frac{1}{a + \frac{1}{a + \frac{1}{a + \frac{1}{...}}}} = 2\).
ie. \(a + \frac{1}{a + \frac{1}{a + \frac{1}{a + \frac{1}{...}}}} = 2\).
Find \(b\) such that \(b+(b+B)^{\frac{1}{2}}=2\), where \(B=(b+B)^{\frac{1}{2}}\).
ie. \(b + \sqrt{b + \sqrt{b + \sqrt{b + \sqrt{...}}}} = 2\).
ie. \(b + \sqrt{b + \sqrt{b + \sqrt{b + \sqrt{...}}}} = 2\).
Find \(c\) such that \(c+(c+C)^{2}=2\), where \(C=(c+C)^{2}\).
In terms of \(k\), find \(d\) such that \(d+(d+D)^{k}=2\), where \(D=(d+D)^{k}\).
Reverse bases
Find two digits \(a\) and \(b\) such that \(ab\) in base 10 is equal to \(ba\) in base 4.
Find two digits \(c\) and \(d\) such that \(cd\) in base 10 is equal to \(dc\) in base 7.
Find two digits \(e\) and \(f\) such that \(ef\) in base 9 is equal to \(fe\) in base 5.
Ninety nine
Source: UKMT Senior Maths Challenge 2013
In a 'ninety nine' shop, all items cost a number of pounds and 99 pence. Susanna spent £65.76. How many items did she buy?
Triangle numbers
Source: ATM Mathematics Teaching 239
Let \(T_n\) be the \(n^\mathrm{th}\) triangle number. Find \(n\) such that: $$T_n+T_{n+1}+T_{n+2}+T_{n+3}=T_{n+4}+T_{n+5}$$
Odd sums
What is \(\frac{1+3}{5+7}\)?
What is \(\frac{1+3+5}{7+9+11}\)?
What is \(\frac{1+3+5+7}{9+11+13+15}\)?
What is \(\frac{1+3+5+7+9}{11+13+15+17+19}\)?
What is \(\frac{\mathrm{sum\ of\ the\ first\ }n\mathrm{\ odd\ numbers}}{\mathrm{sum\ of\ the\ next\ }n\mathrm{\ odd\ numbers}}\)?
x to the power of x
If \(x^{x^{x^{x^{...}}}}\) [\(x\) to the power of (\(x\) to the power of (\(x\) to the power of (\(x\) to the power of ...))) with an infinite number of \(x\)s] is equal to 2, what is the value of \(x\)?