{"id":480,"date":"2011-03-20T19:49:00","date_gmt":"2011-03-20T17:49:00","guid":{"rendered":"https:\/\/www.fussylogic.co.uk\/blog\/?p=480"},"modified":"2012-08-31T09:23:14","modified_gmt":"2012-08-31T08:23:14","slug":"weights-and-measures","status":"publish","type":"post","link":"https:\/\/www.fussylogic.co.uk\/blog\/?p=480","title":{"rendered":"Weights and Measures"},"content":{"rendered":"<blockquote>\n<p>You have a set of balance scales and some weights. You need to be able to measure every whole number weight from 1 to 40 kilograms of (say) flour. What is the smallest number of weights you need and what are their values?<\/p>\n<\/blockquote>\n<p>The first trick is of course that you can put weights on either side, so you have to notice that you don\u00e2\u20ac\u2122t necessarily need to be able to sum to every value, you can subtract as well. If we put the flour on the right hand side, then anything we put on the left adds to the balance, and anything we put on the right subtracts from the balance. For example, if we had two weights, 1kg and 3kg we could make.<\/p>\n<ul>\n<li>1kg<\/li>\n<li>3kg<\/li>\n<li>1kg + 3kg = 4kg<\/li>\n<li>1kg &#8211; 3kg = \u00e2\u20ac\u201c2kg<\/li>\n<li>3kg &#8211; 1kg = 2kg<\/li>\n<\/ul>\n<p>Obviously the negative weight isn\u00e2\u20ac\u2122t going to be useful to us, as we will never be called upon to weigh \u00e2\u20ac\u201c2kg of flour.<\/p>\n<p>So, each weight can be in one of three places:<\/p>\n<ul>\n<li>The same side as the flour<\/li>\n<li>Not the same side as the flour<\/li>\n<li>Not on the scale at all<\/li>\n<\/ul>\n<p>We\u00e2\u20ac\u2122ll call the side the flour is on the \u00e2\u20ac\u0153minus-side\u00e2\u20ac\u009d (since, from the point of view of the plus side, it effectively lowers the weight of the plus side) and the other side the \u00e2\u20ac\u0153plus-side\u00e2\u20ac\u009d.<\/p>\n<p>We\u00e2\u20ac\u2122ll start with two weights, <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Y<\/mi><\/mrow><\/math>\n<p> and <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Z<\/mi><\/mrow><\/math>\n<p>, here are all possible configurations:<\/p>\n<ul>\n<li>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mn>0<\/mn><\/mrow><\/math>\n<\/li>\n<li>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mo>+<\/mo><mi>Y<\/mi><mo>&#8211;<\/mo><mn>0<\/mn><\/mrow><\/math>\n<\/li>\n<li>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mo>+<\/mo><mi>Z<\/mi><mo>&#8211;<\/mo><mn>0<\/mn><\/mrow><\/math>\n<\/li>\n<li>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mn>0<\/mn><mo>&#8211;<\/mo><mi>Y<\/mi><\/mrow><\/math>\n<\/li>\n<li>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mn>0<\/mn><mo>&#8211;<\/mo><mi>Z<\/mi><\/mrow><\/math>\n<\/li>\n<li>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Y<\/mi><mo>&#8211;<\/mo><mi>Z<\/mi><\/mrow><\/math>\n<\/li>\n<li>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Z<\/mi><mo>&#8211;<\/mo><mi>Y<\/mi><\/mrow><\/math>\n<\/li>\n<li>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mo>+<\/mo><mi>Y<\/mi><mo>+<\/mo><mi>Z<\/mi><mo>&#8211;<\/mo><mn>0<\/mn><\/mrow><\/math>\n<\/li>\n<li>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mn>0<\/mn><mo>&#8211;<\/mo><mi>Y<\/mi><mo>&#8211;<\/mo><mi>Z<\/mi><\/mrow><\/math>\n<\/li>\n<\/ul>\n<p>There are nine configurations, which shouldn\u00e2\u20ac\u2122t surprise us. Two weights each in any of three places, is<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mn>3<\/mn><mo>&#215;<\/mo><mn>3<\/mn><mo>=<\/mo><mn>9<\/mn><\/mrow><\/math>\n<\/p>\n<p>Let\u00e2\u20ac\u2122s also recall that we can\u00e2\u20ac\u2122t measure 0kg and we can\u00e2\u20ac\u2122t measure negative weights of flour (since they cannot exist), therefore our equation for real combinations measurable with <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>m<\/mi><\/mrow><\/math>\n<p> weights is<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>N<\/mi><mo stretchy=\"false\">(<\/mo><mi>m<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><mo stretchy=\"false\">(<\/mo><msup><mn>3<\/mn><mi>m<\/mi><\/msup><mo>&#8211;<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><mn>2<\/mn><\/mfrac><\/mrow><\/math>\n<\/p>\n<p>We can therefore be certain then that two weights isn\u00e2\u20ac\u2122t sufficient to measure 40 different weights of flour. Similarly, three weights would be<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mn>3<\/mn><mo>&#215;<\/mo><mn>3<\/mn><mo>&#215;<\/mo><mn>3<\/mn><mo>=<\/mo><mn>27<\/mn><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>N<\/mi><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><mo stretchy=\"false\">(<\/mo><mn>27<\/mn><mo>&#8211;<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><mn>2<\/mn><\/mfrac><mo>=<\/mo><mn>13<\/mn><\/mrow><\/math>\n<\/p>\n<p>Thirteen weights would be insufficient to distinguish between 40 different inputs.<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mn>3<\/mn><mo>&#215;<\/mo><mn>3<\/mn><mo>&#215;<\/mo><mn>3<\/mn><mo>&#215;<\/mo><mn>3<\/mn><mo>=<\/mo><mn>81<\/mn><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>N<\/mi><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><mo stretchy=\"false\">(<\/mo><mn>81<\/mn><mo>&#8211;<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><mn>2<\/mn><\/mfrac><mo>=<\/mo><mn>40<\/mn><\/mrow><\/math>\n<\/p>\n<p>Perfect! What we want is possible. Let\u00e2\u20ac\u2122s list all the configurations of four weights (zero and negatives included). In the following table, \u00e2\u20ac\u01530\u00e2\u20ac\u009d means not on the scales, \u00e2\u20ac\u0153-\u00e2\u20ac\u009d means on the same side as the flour and \u00e2\u20ac\u0153+\u00e2\u20ac\u009d means not on the same side as the flour (the eagle eyed amongst you will notice that this is simply a base\u00e2\u20ac\u201c3 count):<\/p>\n<pre><code>W   X   Y   Z       W   X   Y   Z       W   X   Y   Z\n-------------       -------------       -------------\n+   +   +   +       -   +   +   +       0   +   +   +\n+   +   +   -       -   +   +   -       0   +   +   -\n+   +   +   0       -   +   +   0       0   +   +   0\n+   +   -   +       -   +   -   +       0   +   -   +\n+   +   -   -       -   +   -   -       0   +   -   -\n+   +   -   0       -   +   -   0       0   +   -   0\n+   +   0   +       -   +   0   +       0   +   0   +\n+   +   0   -       -   +   0   -       0   +   0   -\n+   +   0   0       -   +   0   0       0   +   0   0\n+   -   +   +       -   -   +   +       0   -   +   +\n+   -   +   -       -   -   +   -       0   -   +   -\n+   -   +   0       -   -   +   0       0   -   +   0\n+   -   -   +       -   -   -   +       0   -   -   +\n+   -   -   -       -   -   -   -       0   -   -   -\n+   -   -   0       -   -   -   0       0   -   -   0\n+   -   0   +       -   -   0   +       0   -   0   +\n+   -   0   -       -   -   0   -       0   -   0   -\n+   -   0   0       -   -   0   0       0   -   0   0\n+   0   +   +       -   0   +   +       0   0   +   +\n+   0   +   -       -   0   +   -       0   0   +   -\n+   0   +   0       -   0   +   0       0   0   +   0\n+   0   -   +       -   0   -   +       0   0   -   +\n+   0   -   -       -   0   -   -       0   0   -   -\n+   0   -   0       -   0   -   0       0   0   -   0\n+   0   0   +       -   0   0   +       0   0   0   +\n+   0   0   -       -   0   0   -       0   0   0   -\n+   0   0   0       -   0   0   0       0   0   0   0\n<\/code><\/pre>\n<p>Let us (for our own convenience) assume that <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><\/mrow><\/math>\n<p> is the heaviest, running to <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Z<\/mi><\/mrow><\/math>\n<p>, the lightest. Such that<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo>&#8805;<\/mo><mi>X<\/mi><mo>&#8805;<\/mo><mi>Y<\/mi><mo>&#8805;<\/mo><mi>Z<\/mi><\/mrow><\/math>\n<\/p>\n<p>We know that the most we can put on one side is all the weights. Therefore all of our weights must add up to some maximum, <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>N<\/mi><\/mrow><\/math>\n<p>.<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo>+<\/mo><mi>X<\/mi><mo>+<\/mo><mi>Y<\/mi><mo>+<\/mo><mi>Z<\/mi><mo>=<\/mo><mi>N<\/mi><\/mrow><\/math>\n<\/p>\n<p>The second heaviest weight we are asked to weigh is <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>N<\/mi><mo>&#8211;<\/mo><mn>1<\/mn><\/mrow><\/math>\n<p>, and the least we can take off is the smallest weight, <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Z<\/mi><\/mrow><\/math>\n<p>, therefore<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo>+<\/mo><mi>X<\/mi><mo>+<\/mo><mi>Y<\/mi><mo>=<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>1<\/mn><\/mrow><\/math>\n<\/p>\n<p>Therefore <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Z<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><\/math>\n<p> (subtract these two equations). Which then enables us to measure <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>N<\/mi><mo>&#8211;<\/mo><mn>2<\/mn><\/mrow><\/math>\n<p> by putting <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Z<\/mi><\/mrow><\/math>\n<p> on the minus-side.<\/p>\n<p>Now we must measure <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>N<\/mi><mo>&#8211;<\/mo><mn>3<\/mn><\/mrow><\/math>\n<p>. The least we can do is replace <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Z<\/mi><\/mrow><\/math>\n<p> on the plus-side and remove <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Y<\/mi><\/mrow><\/math>\n<p>.<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo>+<\/mo><mi>X<\/mi><mo>+<\/mo><mi>Z<\/mi><mo>=<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>3<\/mn><\/mrow><\/math>\n<\/p>\n<p>Therefore <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Y<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><\/math>\n<p>. Which then enables us to measure <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mo stretchy=\"false\">(<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/math>\n<p>, <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mo stretchy=\"false\">(<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/math>\n<p>, <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mo stretchy=\"false\">(<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>6<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/math>\n<p>, <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mo stretchy=\"false\">(<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>7<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/math>\n<p>, and <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mo stretchy=\"false\">(<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>8<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/math>\n<p>. Recapping:<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo>+<\/mo><mi>X<\/mi><mo>+<\/mo><mi>Y<\/mi><mo>+<\/mo><mi>Z<\/mi><mo>=<\/mo><mi>N<\/mi><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo>+<\/mo><mi>X<\/mi><mo>+<\/mo><mi>Y<\/mi><mo>+<\/mo><mn>0<\/mn><mo>=<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>1<\/mn><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo>+<\/mo><mi>X<\/mi><mo>+<\/mo><mi>Y<\/mi><mo>&#8211;<\/mo><mi>Z<\/mi><mo>=<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>2<\/mn><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo>+<\/mo><mi>X<\/mi><mo>+<\/mo><mn>0<\/mn><mo>+<\/mo><mi>Z<\/mi><mo>=<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>3<\/mn><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo>+<\/mo><mi>X<\/mi><mo>+<\/mo><mn>0<\/mn><mo>+<\/mo><mn>0<\/mn><mo>=<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>4<\/mn><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo>+<\/mo><mi>X<\/mi><mo>+<\/mo><mn>0<\/mn><mo>&#8211;<\/mo><mi>Z<\/mi><mo>=<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>5<\/mn><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo>+<\/mo><mi>X<\/mi><mo>&#8211;<\/mo><mi>Y<\/mi><mo>+<\/mo><mi>Z<\/mi><mo>=<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>6<\/mn><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo>+<\/mo><mi>X<\/mi><mo>&#8211;<\/mo><mi>Y<\/mi><mo>+<\/mo><mn>0<\/mn><mo>=<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>7<\/mn><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo>+<\/mo><mi>X<\/mi><mo>&#8211;<\/mo><mi>Y<\/mi><mo>&#8211;<\/mo><mi>Z<\/mi><mo>=<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>8<\/mn><\/mrow><\/math>\n<\/p>\n<p>You should be able to see the pattern from the table evolving now. Each weight is either positive, off, or negative. Which means we get double duty from each one. Taking <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Z<\/mi><\/mrow><\/math>\n<p> off lowers the total by 1kg; adding <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Z<\/mi><\/mrow><\/math>\n<p> to the negative side lowers the total by another 1kg. So, with <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Z<\/mi><\/mrow><\/math>\n<p> we can subtract 0, 1 or 2 from a given point (where <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Z<\/mi><\/mrow><\/math>\n<p> is \u00e2\u20ac\u0153on\u00e2\u20ac\u009d). <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Y<\/mi><\/mrow><\/math>\n<p> lets us subtract 0, 3 or 6 from a given point (where <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Y<\/mi><\/mrow><\/math>\n<p> is \u00e2\u20ac\u0153on\u00e2\u20ac\u009d). Combined <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Z<\/mi><\/mrow><\/math>\n<p> and <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Y<\/mi><\/mrow><\/math>\n<p> lets us subtract 0, 1, 2, 3, 4, 5, 6, 7, or 8 from a given point (where <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Y<\/mi><\/mrow><\/math>\n<p> and <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Z<\/mi><\/mrow><\/math>\n<p> are \u00e2\u20ac\u0153on\u00e2\u20ac\u009d).<\/p>\n<p>It\u00e2\u20ac\u2122s now pretty simple to carry on. The next smallest change we can make is to remove the second heaviest weight; we want that to give us <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mo stretchy=\"false\">(<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>9<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/math>\n<p>:<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo>+<\/mo><mn>0<\/mn><mo>+<\/mo><mi>Y<\/mi><mo>+<\/mo><mi>Z<\/mi><mo>=<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>9<\/mn><\/mrow><\/math>\n<\/p>\n<p>Therefore <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>X<\/mi><mo>=<\/mo><mn>9<\/mn><\/mrow><\/math>\n<p>. <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Y<\/mi><\/mrow><\/math>\n<p> and <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Z<\/mi><\/mrow><\/math>\n<p> are both \u00e2\u20ac\u0153on\u00e2\u20ac\u009d so we know that we can subtract another 8 kg from <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mo stretchy=\"false\">(<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>9<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/math>\n<p> in sequence, taking us to:<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo>+<\/mo><mn>0<\/mn><mo>+<\/mo><mi>Y<\/mi><mo>+<\/mo><mn>0<\/mn><mo>=<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>10<\/mn><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo>+<\/mo><mn>0<\/mn><mo>+<\/mo><mi>Y<\/mi><mo>&#8211;<\/mo><mi>Z<\/mi><mo>=<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>11<\/mn><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo>+<\/mo><mn>0<\/mn><mo>+<\/mo><mn>0<\/mn><mo>+<\/mo><mi>Z<\/mi><mo>=<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>12<\/mn><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo>+<\/mo><mn>0<\/mn><mo>+<\/mo><mn>0<\/mn><mo>+<\/mo><mn>0<\/mn><mo>=<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>13<\/mn><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo>+<\/mo><mn>0<\/mn><mo>+<\/mo><mn>0<\/mn><mo>&#8211;<\/mo><mi>Z<\/mi><mo>=<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>14<\/mn><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo>+<\/mo><mn>0<\/mn><mo>&#8211;<\/mo><mi>Y<\/mi><mo>+<\/mo><mi>Z<\/mi><mo>=<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>15<\/mn><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo>+<\/mo><mn>0<\/mn><mo>&#8211;<\/mo><mi>Y<\/mi><mo>+<\/mo><mn>0<\/mn><mo>=<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>16<\/mn><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo>+<\/mo><mn>0<\/mn><mo>&#8211;<\/mo><mi>Y<\/mi><mo>&#8211;<\/mo><mi>Z<\/mi><mo>=<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>17<\/mn><\/mrow><\/math>\n<\/p>\n<p>Now we have to put X on the minus side, from there we know that <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Y<\/mi><\/mrow><\/math>\n<p> and <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Z<\/mi><\/mrow><\/math>\n<p> back on will give us the ability to do another 8 kg:<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo>&#8211;<\/mo><mi>X<\/mi><mo>+<\/mo><mi>Y<\/mi><mo>+<\/mo><mi>Z<\/mi><mo>=<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>18<\/mn><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mo>&#8230;<\/mo><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo>&#8211;<\/mo><mi>X<\/mi><mo>&#8211;<\/mo><mi>Y<\/mi><mo>&#8211;<\/mo><mi>Z<\/mi><mo>=<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>26<\/mn><\/mrow><\/math>\n<\/p>\n<p>That only leaves us with the option of putting <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>X<\/mi><\/mrow><\/math>\n<p>, <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Y<\/mi><\/mrow><\/math>\n<p> and <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Z<\/mi><\/mrow><\/math>\n<p> back on, and taking <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><\/mrow><\/math>\n<p> off, which we want to give us the next value, <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>N<\/mi><mo>&#8211;<\/mo><mn>27<\/mn><\/mrow><\/math>\n<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mn>0<\/mn><mo>+<\/mo><mi>X<\/mi><mo>+<\/mo><mi>Y<\/mi><mo>+<\/mo><mi>Z<\/mi><mo>=<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>27<\/mn><\/mrow><\/math>\n<\/p>\n<p>Subtracting this from the initial equation, <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mo stretchy=\"false\">(<\/mo><mi>W<\/mi><mo>+<\/mo><mi>X<\/mi><mo>+<\/mo><mi>Y<\/mi><mo>+<\/mo><mi>Z<\/mi><mo>=<\/mo><mi>N<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/math>\n<p>:<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo>=<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mo stretchy=\"false\">(<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>27<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mo>=<\/mo><mn>27<\/mn><\/mrow><\/math>\n<\/p>\n<p>So, <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo>=<\/mo><mn>27<\/mn><\/mrow><\/math>\n<p>. Further, <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><\/mrow><\/math>\n<p>, <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>X<\/mi><\/mrow><\/math>\n<p>, <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Y<\/mi><\/mrow><\/math>\n<p> and <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Z<\/mi><\/mrow><\/math>\n<p> are now known, so we can calculate that <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>N<\/mi><mo>=<\/mo><mn>40<\/mn><\/mrow><\/math>\n<p>.<\/p>\n<p>We already know that we can position <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>X<\/mi><\/mrow><\/math>\n<p>, <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Y<\/mi><\/mrow><\/math>\n<p> and <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Z<\/mi><\/mrow><\/math>\n<p> to subtract up to 26, which is just what we need to get us to 1kg.<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>N<\/mi><mo>=<\/mo><mn>40<\/mn><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo>=<\/mo><mn>27<\/mn><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>X<\/mi><mo>=<\/mo><mn>9<\/mn><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Y<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Z<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><\/math>\n<\/p>\n<p>QED.<\/p>\n<hr \/>\n<p>Now let\u00e2\u20ac\u2122s go even more general. For <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>n<\/mi><\/mrow><\/math>\n<p> weights, what is the maximum weight of flour we can weigh? What weights do we need?<\/p>\n<p>We\u00e2\u20ac\u2122ll plough through this a little faster, first let\u00e2\u20ac\u2122s add an additional weight:<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>V<\/mi><mo>+<\/mo><mi>W<\/mi><mo>+<\/mo><mi>X<\/mi><mo>+<\/mo><mi>Y<\/mi><mo>+<\/mo><mi>Z<\/mi><mo>=<\/mo><mi>N<\/mi><\/mrow><\/math>\n<\/p>\n<p>We already know what <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><\/mrow><\/math>\n<p>, <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>X<\/mi><\/mrow><\/math>\n<p>, <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Y<\/mi><\/mrow><\/math>\n<p> and <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Z<\/mi><\/mrow><\/math>\n<p> are, their derivation is utterly unchanged by the presence of <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>V<\/mi><\/mrow><\/math>\n<p>. We run out when we come to:<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>V<\/mi><mo>&#8211;<\/mo><mi>W<\/mi><mo>&#8211;<\/mo><mi>X<\/mi><mo>&#8211;<\/mo><mi>Y<\/mi><mo>&#8211;<\/mo><mi>Z<\/mi><mo>=<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>80<\/mn><\/mrow><\/math>\n<\/p>\n<p>I know that it\u00e2\u20ac\u2122s <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>N<\/mi><mo>&#8211;<\/mo><mn>80<\/mn><\/mrow><\/math>\n<p> as follows:<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>V<\/mi><mo>+<\/mo><mi>W<\/mi><mo>+<\/mo><mi>X<\/mi><mo>+<\/mo><mi>Y<\/mi><mo>+<\/mo><mi>Z<\/mi><mo>=<\/mo><mi>N<\/mi><\/mrow><\/math>\n<p> (eq1)  <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo>+<\/mo><mi>X<\/mi><mo>+<\/mo><mi>Y<\/mi><mo>+<\/mo><mi>Z<\/mi><mo>=<\/mo><mn>40<\/mn><\/mrow><\/math>\n<p> (eq2)  <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>V<\/mi><mo>&#8211;<\/mo><mi>W<\/mi><mo>&#8211;<\/mo><mi>X<\/mi><mo>&#8211;<\/mo><mi>Y<\/mi><mo>&#8211;<\/mo><mi>Z<\/mi><mo>=<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>80<\/mn><\/mrow><\/math>\n<p> (eq1)-<\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mn>2<\/mn><mo>&#215;<\/mo><\/mrow><\/math>\n<p>(eq2)<\/p>\n<p>As before, we put all <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><\/mrow><\/math>\n<p>, <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>X<\/mi><\/mrow><\/math>\n<p>, <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Y<\/mi><\/mrow><\/math>\n<p> and <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>Z<\/mi><\/mrow><\/math>\n<p> back on the plus side, and take <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>V<\/mi><\/mrow><\/math>\n<p> off, and that must be equal to our next integer weight:<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mn>0<\/mn><mo>+<\/mo><mi>W<\/mi><mo>+<\/mo><mi>X<\/mi><mo>+<\/mo><mi>Y<\/mi><mo>+<\/mo><mi>Z<\/mi><mo>=<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>81<\/mn><\/mrow><\/math>\n<\/p>\n<p>As before, we subtract this from our initial equation, <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>V<\/mi><mo>+<\/mo><mi>W<\/mi><mo>+<\/mo><mi>X<\/mi><mo>+<\/mo><mi>Y<\/mi><mo>+<\/mo><mi>Z<\/mi><mo>=<\/mo><mi>N<\/mi><\/mrow><\/math>\n<p>:<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>V<\/mi><mo>+<\/mo><mi>W<\/mi><mo>&#8211;<\/mo><mi>W<\/mi><mo>+<\/mo><mi>X<\/mi><mo>&#8211;<\/mo><mi>X<\/mi><mo>+<\/mo><mi>Y<\/mi><mo>&#8211;<\/mo><mi>Y<\/mi><mo>+<\/mo><mi>Z<\/mi><mo>&#8211;<\/mo><mi>Z<\/mi><mo>=<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mo stretchy=\"false\">(<\/mo><mi>N<\/mi><mo>&#8211;<\/mo><mn>81<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>V<\/mi><mo>=<\/mo><mn>81<\/mn><\/mrow><\/math>\n<\/p>\n<p>We shouldn\u00e2\u20ac\u2122t be surprised; as we said, each weight can be used twice, once when we take it off the plus side, and once when we add it to the minus side. Therefore, the nth weight is the sum of all the previous weights multiplied by two, plus one to get us to the next unknown.<\/p>\n<p>Now we\u00e2\u20ac\u2122ll label the <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>n<\/mi><\/mrow><\/math>\n<p>th weight <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/math>\n<p>, with <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>n<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><\/math>\n<p> being the first weight, and <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/math>\n<p> equal to one. <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/math>\n<p> can be expressed as a function of the sum of all the previous weights:<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mrow><mo stretchy=\"true\">(<\/mo><munderover><mo>&#8721;<\/mo><mrow><mi>i<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><mo>&#8211;<\/mo><mn>1<\/mn><\/mrow><\/munderover><mrow><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mo stretchy=\"true\">)<\/mo><\/mrow><mo>&#215;<\/mo><mn>2<\/mn><mo>+<\/mo><mn>1<\/mn><\/mrow><\/math>\n<\/p>\n<p>So:<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>1<\/mn><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><mo>&#215;<\/mo><mn>2<\/mn><mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mn>3<\/mn><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><mo>&#215;<\/mo><mn>2<\/mn><mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mn>9<\/mn><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><mo>&#215;<\/mo><mn>2<\/mn><mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mn>27<\/mn><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><mo>&#215;<\/mo><mn>2<\/mn><mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mn>81<\/mn><\/mrow><\/math>\n<\/p>\n<p>We should note that the maximum we can weigh with <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>n<\/mi><\/mrow><\/math>\n<p> weights is the sum of those <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>n<\/mi><\/mrow><\/math>\n<p> weights, or<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>N<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><munderover><mo>&#8721;<\/mo><mrow><mi>i<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><mi>n<\/mi><\/munderover><mrow><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mi>i<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mrow><\/math>\n<\/p>\n<p>Therefore,<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>N<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>&#8211;<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>&#215;<\/mo><mn>2<\/mn><mo>+<\/mo><mn>1<\/mn><\/mrow><\/math>\n<\/p>\n<p>We should also note that we can get rid of the summation, and simply define <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>N<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/math>\n<p> in terms of <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>N<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>&#8211;<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/math>\n<p>:<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>N<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>N<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>&#8211;<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/math>\n<\/p>\n<p>Substituting for <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/math>\n<p>:<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>N<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>N<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>&#8211;<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mi>N<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>&#8211;<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>&#215;<\/mo><mn>2<\/mn><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mo>=<\/mo><mn>3<\/mn><mi>N<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>&#8211;<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>1<\/mn><\/mrow><\/math>\n<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>N<\/mi><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/math>\n<p> we have to define as 1 to match our <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/math>\n<p> defined as 1.<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>N<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mrow><mn>1<\/mn><mo>,<\/mo><mn>4<\/mn><mo>,<\/mo><mn>13<\/mn><mo>,<\/mo><mn>40<\/mn><mo>,<\/mo><mn>121<\/mn><mo>,<\/mo><mn>364<\/mn><mo>,<\/mo><mo>&#8230;<\/mo><\/mrow><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mrow><mn>1<\/mn><mo>,<\/mo><mn>3<\/mn><mo>,<\/mo><mn>9<\/mn><mo>,<\/mo><mn>27<\/mn><mo>,<\/mo><mn>81<\/mn><mo>,<\/mo><mn>243<\/mn><mo>,<\/mo><mn>729<\/mn><mo>,<\/mo><mo>&#8230;<\/mo><\/mrow><\/mrow><\/math>\n<\/p>\n<p>We\u00e2\u20ac\u2122re still defining each term in terms of the previous term, which we needn\u00e2\u20ac\u2122t do. We\u00e2\u20ac\u2122ve already got an equation for <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>N<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/math>\n<p>:<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>N<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><mo stretchy=\"false\">(<\/mo><msup><mn>3<\/mn><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msup><mo>&#8211;<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><mn>2<\/mn><\/mfrac><\/mrow><\/math>\n<\/p>\n<p>This is the very first equation I gave, but adjusted for a start-from-zero <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>n<\/mi><\/mrow><\/math>\n<p>, instead of start-from-one, <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>m<\/mi><\/mrow><\/math>\n<p>. Substituting this into our equation for <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><\/mrow><\/math>\n<p>:<\/p>\n<p><math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mi>W<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mfrac><mrow><msup><mn>3<\/mn><mrow><mi>n<\/mi><mo>&#8211;<\/mo><mn>1<\/mn><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msup><mo>&#8211;<\/mo><mn>1<\/mn><\/mrow><mn>2<\/mn><\/mfrac><mo stretchy=\"false\">)<\/mo><mo>&#215;<\/mo><mn>2<\/mn><mo>+<\/mo><mn>1<\/mn><\/mrow><\/math>\n<p> <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><mo>=<\/mo><msup><mn>3<\/mn><mi>n<\/mi><\/msup><\/mrow><\/math>\n<\/p>\n<p>This shouldn\u00e2\u20ac\u2122t be a surprise to us really. The weights are a base\u00e2\u20ac\u201c3 counting system, for any base, the nth digit\u00e2\u20ac\u2122s multiplier is given by <\/p>\n<math display=\"inline\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow><msup><mi>B<\/mi><mi>n<\/mi><\/msup><\/mrow><\/math>\n<p>. If we\u00e2\u20ac\u2122d been really clever, we would have recognised this right from the start and just gone straight to the answer. Oh well.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>You have a set of balance scales and some weights. You need to be able to measure every whole number weight from 1 to 40 kilograms of (say) flour. What is the smallest number of weights you need and what are their values? The first trick is of course that you can put weights on\u2026 <span class=\"read-more\"><a href=\"https:\/\/www.fussylogic.co.uk\/blog\/?p=480\">Read More &raquo;<\/a><\/span><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[1],"tags":[32,33],"_links":{"self":[{"href":"https:\/\/www.fussylogic.co.uk\/blog\/index.php?rest_route=\/wp\/v2\/posts\/480"}],"collection":[{"href":"https:\/\/www.fussylogic.co.uk\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.fussylogic.co.uk\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.fussylogic.co.uk\/blog\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.fussylogic.co.uk\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=480"}],"version-history":[{"count":6,"href":"https:\/\/www.fussylogic.co.uk\/blog\/index.php?rest_route=\/wp\/v2\/posts\/480\/revisions"}],"predecessor-version":[{"id":596,"href":"https:\/\/www.fussylogic.co.uk\/blog\/index.php?rest_route=\/wp\/v2\/posts\/480\/revisions\/596"}],"wp:attachment":[{"href":"https:\/\/www.fussylogic.co.uk\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=480"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.fussylogic.co.uk\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=480"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.fussylogic.co.uk\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=480"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}