writers were clearly looking for a riddle that most of the audience would be able to work out or at least understand for themselves. But it’s an age-old arithmetical problem based on the subtraction of relative prime numbers. Relative prime numbers are numbers where the only whole number that both can be divided by is 1. That doesn’t mean that either is a true prime number,just that they can both be divided only by 1. Thus 15 and 16, although not primes, are both relative primes, whereas 15 and 21 are not, because they can both be divided by 3.
Euclid found a way to solve relative prime number problems like the
Die Hard
riddle 2,300 years ago. The proof is complex, but the arithmetic relatively simple, and it allows us to solve any similar problem, such as how you would get a 13-minute egg from a 5-minute and a 9-minute egg timer.
(You ‘calibrate’ the 5-minute timer to 4 minutes, by starting them both off together, stopping when the 5-minute timer runs out – leaving 4 minutes on the 9-minute timer – starting both off together again and stopping when the 9-minute timer runs out, leaving just a minute on the 5-minute timer. You can then cook your egg for exactly 13 minutes by starting with the 9-minute, then reversing the 5-minute timer for the remaining 4 minutes.)
All these solutions can be expressed by mathematical equations. If one of the measures you have is
p
and the other is
q
, you can find the solution you need,
k,
with the following equation. In this,
m
is the number of times you need to fill or empty
p,
and
n
is the number of times you need to fill or empty
q
.
mp
+
nq
=
k
If
m
or
n
is negative, it means emptying the jug (or egg timer); if
m
or
n
is positive it means filling the jug. In thecase of the
Die Hard
jugs,
p
is 3 and
q
is 5. So you would get
k
is 4 if
m
is plus 3 and
n
minus 1.
3 × 3 + –1 × 5 = 4
That means filling the 3-gallon jug three times and emptying the 5-gallon jug once. Alternatively, you could go with
m
as minus 2 and
n
as plus 2, i.e. emptying the 3-gallon jug twice and filling the 5-gallon jug twice.
–2 × 3 + 2 × 5 = 4
Both of these are different solutions from John McCain’s and require a third large jug as a reservoir, but they are equally valid solutions.
Was it fair that a woman’s planning application for painting her door purple in a conservation area was declined?
(Land Economy, Cambridge)
The idea that our past needed protecting by law from the ravages of modernisation first emerged in the mid-nineteenth century. As mass industrialisation and urbanisation swept away age-old ways of life, many believed something valuable was being lost – not just old and often beautiful things but our whole connection to the past, intangible, fragile but immensely precious. It’s no coincidence that the first great strides into the modern urban, industrial worldcoincided with the revival of the Gothic style of architecture, the medievalism of the pre-Raphaelite art movement and the emphasis on traditional skills of the Arts and Crafts movement.
The preservation movement began with the scheduling of Ancient Monuments in 1882 to protect important ancient archaeological and historical sites. In 1947, the realisation that unique old buildings needed protection too against demolition or simply being ruined by modernisation led to the idea of ‘listing’ buildings. Then conservation areas came in during the 1960s; battlefields, historic parks and gardens were placed on registers in the 1980s; and, most recently, marine archaeology has been given protection.
Conservation areas recognise that not just stately homes, ancient castles and quaint medieval cottages are part of our heritage but so too are whole neighbourhoods of ‘ordinary’ historic houses. The idea is to preserve the special historic character of these neighbourhoods. Yet conservation areas cannot be locked into the past like museums where the buildings are simply preserved ‘in aspic’. They are
Euclid found a way to solve relative prime number problems like the
Die Hard
riddle 2,300 years ago. The proof is complex, but the arithmetic relatively simple, and it allows us to solve any similar problem, such as how you would get a 13-minute egg from a 5-minute and a 9-minute egg timer.
(You ‘calibrate’ the 5-minute timer to 4 minutes, by starting them both off together, stopping when the 5-minute timer runs out – leaving 4 minutes on the 9-minute timer – starting both off together again and stopping when the 9-minute timer runs out, leaving just a minute on the 5-minute timer. You can then cook your egg for exactly 13 minutes by starting with the 9-minute, then reversing the 5-minute timer for the remaining 4 minutes.)
All these solutions can be expressed by mathematical equations. If one of the measures you have is
p
and the other is
q
, you can find the solution you need,
k,
with the following equation. In this,
m
is the number of times you need to fill or empty
p,
and
n
is the number of times you need to fill or empty
q
.
mp
+
nq
=
k
If
m
or
n
is negative, it means emptying the jug (or egg timer); if
m
or
n
is positive it means filling the jug. In thecase of the
Die Hard
jugs,
p
is 3 and
q
is 5. So you would get
k
is 4 if
m
is plus 3 and
n
minus 1.
3 × 3 + –1 × 5 = 4
That means filling the 3-gallon jug three times and emptying the 5-gallon jug once. Alternatively, you could go with
m
as minus 2 and
n
as plus 2, i.e. emptying the 3-gallon jug twice and filling the 5-gallon jug twice.
–2 × 3 + 2 × 5 = 4
Both of these are different solutions from John McCain’s and require a third large jug as a reservoir, but they are equally valid solutions.
Was it fair that a woman’s planning application for painting her door purple in a conservation area was declined?
(Land Economy, Cambridge)
The idea that our past needed protecting by law from the ravages of modernisation first emerged in the mid-nineteenth century. As mass industrialisation and urbanisation swept away age-old ways of life, many believed something valuable was being lost – not just old and often beautiful things but our whole connection to the past, intangible, fragile but immensely precious. It’s no coincidence that the first great strides into the modern urban, industrial worldcoincided with the revival of the Gothic style of architecture, the medievalism of the pre-Raphaelite art movement and the emphasis on traditional skills of the Arts and Crafts movement.
The preservation movement began with the scheduling of Ancient Monuments in 1882 to protect important ancient archaeological and historical sites. In 1947, the realisation that unique old buildings needed protection too against demolition or simply being ruined by modernisation led to the idea of ‘listing’ buildings. Then conservation areas came in during the 1960s; battlefields, historic parks and gardens were placed on registers in the 1980s; and, most recently, marine archaeology has been given protection.
Conservation areas recognise that not just stately homes, ancient castles and quaint medieval cottages are part of our heritage but so too are whole neighbourhoods of ‘ordinary’ historic houses. The idea is to preserve the special historic character of these neighbourhoods. Yet conservation areas cannot be locked into the past like museums where the buildings are simply preserved ‘in aspic’. They are
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