diff --git a/Doc/Sd1/statements.xml b/Doc/Sd1/statements.xml
index 889a94c47ff489de6b3ea8dc11352fbe7cd2fcce..1ad7de93a717b4e2789d6d3fb84a78b574957e91 100644
--- a/Doc/Sd1/statements.xml
+++ b/Doc/Sd1/statements.xml
@@ -6640,10 +6640,14 @@ Found (28, 96, 100)</screen>
combinations being evaluated and the number of Pythagorean
triples being found creating a final line like:</para>
- <screen>52 triples found by filtering 131935 combinations</screen>
+ <screen>...
+(60, 63, 87)
+(60, 80, 100)
+(65, 72, 97)
+52 triples found by filtering 34766 combinations</screen>
<para>Your solution should find all 52 triples by filtering
- from at most 131935 combinations. If you do need less
+ from at most 34766 combinations. If you do need less
combinations in the first place please let me know. I'm more
than happy publishing an enhanced solution.</para>
</question>
@@ -6654,67 +6658,92 @@ Found (28, 96, 100)</screen>
<orderedlist>
<listitem>
- <para>We calculate <inlineequation>
- <m:math display="inline">
- <m:msup>
- <m:mi>c</m:mi>
-
- <m:mi>2</m:mi>
- </m:msup>
- </m:math>
- </inlineequation> beforehand:</para>
+ <para>We start with the rectangle's smaller leg:</para>
<programlisting language="java">...
-for (int c = 2; c <= LIMIT; c++) {
- <emphasis role="red">final int cSquare = c * c</emphasis>;
+for (int a = 1; a < LIMIT - 1; a++) {
+
+}
...</programlisting>
+
+ <note>
+ <para>Since <inlineequation>
+ <m:math display="inline">
+ <m:mrow>
+ <m:mi>1</m:mi>
+
+ <m:mo><</m:mo>
+
+ <m:mi>a</m:mi>
+
+ <m:mo>≤</m:mo>
+
+ <m:mi>b</m:mi>
+
+ <m:mo><</m:mo>
+
+ <m:mi>c</m:mi>
+
+ <m:mo>≤</m:mo>
+
+ <m:mi>LIMIT</m:mi>
+ </m:mrow>
+ </m:math>
+ </inlineequation> holds the rectangle's leg a can only
+ exceed <code>LIMIT - 2</code>. </para>
+ </note>
</listitem>
<listitem>
- <para>Within the next inner loop we calculate
- <inlineequation>
+ <para>Since <inlineequation>
<m:math display="inline">
<m:mrow>
- <m:msup>
- <m:mi>c</m:mi>
-
- <m:mi>2</m:mi>
- </m:msup>
-
- <m:mo>-</m:mo>
+ <m:mi>a</m:mi>
- <m:msup>
- <m:mi>b</m:mi>
+ <m:mo>≤</m:mo>
- <m:mi>2</m:mi>
- </m:msup>
+ <m:mi>b</m:mi>
</m:mrow>
</m:math>
- </inlineequation> beforehand as well:</para>
+ </inlineequation> holds the next inner loop is
+ straightforward:</para>
<programlisting language="java">...
-for (int b = 1; b < c; b++) { // b must not exceed c since a == 0 otherwise
- <emphasis role="red">final int cSquare_minus_bSquare = cSquare - b * b</emphasis>;
+for (int b = a; b < LIMIT; b++) {
+ final int /* useful constants*/
+ aPlusB = a + b,
+ aSquarePlusBsquare = a * a + b * b;
+ ...
+}
...</programlisting>
+
+ <note>
+ <para>The two variables <code>aPlusB</code> and
+ <code>aSquarePlusBsquare</code> will not have to be
+ re-calculated inside the innermost loop yet to be
+ defined.</para>
+ </note>
</listitem>
<listitem>
- <para>The innermost loop will terminate on two
- conditions:</para>
+ <para>We are dealing with triangles. c must thus be longer
+ than the triangle's longest leg. The innermost loop will
+ terminate on multiple conditions:</para>
<programlisting language="java">...
-for (int a = 1;
- <emphasis role="red">a <= b</emphasis> && // Only ordered triples
- <emphasis role="red">a * a <= cSquare_minus_bSquare</emphasis>; // Only till square root of c² - b²
- a++) { ...</programlisting>
+for (int c = b + 1;
+ <emphasis role="red">c <= LIMIT</emphasis> &&
+ <emphasis role="red">c < aPlusB</emphasis> && //
+ <emphasis role="red">c * c <= aSquarePlusBsquare</emphasis>;
+ c++) { ...</programlisting>
</listitem>
<listitem>
<para>Finally we re-write our filtering condition:</para>
<programlisting language="java">...
-if (<emphasis role="red">a * a == cSquare_minus_bSquare</emphasis>) { // Equivalent to a² + b² == c²
- System.out.println("Found (" + a + ", " + b + ", " + c + ")");
+if (<emphasis role="red">aSquarePlusBsquare == c * c</emphasis>) { // Equivalent to a² + b² == c²
+ System.out.format("(%2d, %2d, %3d)\n", a, b , c);
}</programlisting>
</listitem>
</orderedlist>
@@ -6728,24 +6757,27 @@ if (<emphasis role="red">a * a == cSquare_minus_bSquare</emphasis>) {
int comparisons = 0, countTriples = 0;
-for (int c = 2; c <= LIMIT; c++) {
- final int cSquare = c * c;
- for (int b = 1; b < c; b++) { // b must not exceed c since a == 0 otherwise
- final int cSquare_minus_bSquare = cSquare - b * b; // c² - b²
- for (int a = 1;
- a <= b && // Only ordered triples
- a * a <= cSquare_minus_bSquare; // Only till square root of c² - b²
- a++) {
- if (a * a == cSquare_minus_bSquare ) { // Equivalent to a² + b² == c²
- System.out.println(
- "Found (" + a + ", " + b + ", " + c + ")");
- countTriples++;
- }
- comparisons++;
+for (int a = 1; a < LIMIT - 1; a++) {
+ for (int b = a; b < LIMIT; b++) {
+ final int
+ aPlusB = a + b,
+ aSquarePlusBsquare = a * a + b * b;
+
+ for (int c = b + 1; // c must be longer than the maximum of a and b
+ c <= LIMIT &&
+ c < aPlusB && // c must be smaller than the sum of a and b
+ c * c <= aSquarePlusBsquare; // c² must not be larger than a² + b²
+ c++) {
+ if (aSquarePlusBsquare == c * c) { // Equivalent to a² + b² == c²
+ System.out.format("(%2d, %2d, %3d)\n", a, b , c);
+ countTriples++;
}
+ comparisons++;
}
+ }
}
-System.out.println(countTriples + " triples found by filtering " + comparisons + " combinations");</programlisting>
+System.out.println(countTriples + " triples found by filtering " + comparisons + " combinations");
+</programlisting>
</answer>
</qandaentry>
</qandadiv>