Descartes Sangaku : The Aberration

• Getting the radii of the circles one after the other is very easy using Descartes circle's theorem, the real difficulty is finding a general expression to compute the radius of the nth circle

• $\color{#69047E}Purple$ $\color{#69047E}circles$ : The radii of the circles are in the sequence : $\boxed{\frac{3600}{481}}$ , $\boxed{\frac{3600}{724}}$ , $\boxed{\frac{3600}{1129}}$ , $\boxed{\frac{3600}{1696}}$ , $\boxed{\frac{3600}{2425}}$
• The numerator never changes and stay $3600$ but the denominator is trickier but we notice that:
• $\boxed{724-481=243=81+162\cdot 1}$
• $\boxed{1169-724=405=81+162\cdot 2}$
• $\boxed{1696-1129=567=81+162\cdot 3}$
• $\boxed{2425-1696=729=81+162\cdot 4}$
• We can conclude that : $\boxed{Rp_n=\frac{3600}{\:400+81n+162n\cdot \:\frac{n-1}{2}}=\frac{3600}{\:81n^2+400}}$
• Then $\boxed{Rp_{100}=\frac{9}{2026}}$ then $\boxed{a=9}$ and $\boxed{b=2026}$

• $\color{#20A900}Green$ $\color{#20A900}circles$ : The radii of the circles are in the sequence : $\boxed{\frac{900}{589}}$ , $\boxed{\frac{900}{751}}$ , $\boxed{\frac{900}{1075}}$ , $\boxed{\frac{900}{1561}}$ , $\boxed{\frac{900}{2209}}$
• The numerator never changes and stay $900$ but the denominator is trickier but we notice that:
• $\boxed{751-589=162=162\cdot 1}$
• $\boxed{1075-751=324=162\cdot 2}$
• $\boxed{1561-1075=486=162\cdot 3}$
• $\boxed{2209-1561=648=162\cdot 4}$
• We can conclude that : $\boxed{Rg_n=\frac{900}{589+162n\cdot \frac{n-1}{2}}=\frac{900}{81n^2-81n+589}}$

• Then $\boxed{Rg_{100}=\frac{900}{802489}}$ then $\boxed{c=900}$ and $\boxed{d=802489}$

• $\color{#3D99F6}Blue$ $\color{#3D99F6}circles$ : The radii of the circles are in the sequence : $\boxed{\frac{3600}{1249}}$ , $\boxed{\frac{3600}{1897}}$ , $\boxed{\frac{3600}{3193}}$ , $\boxed{\frac{3600}{5137}}$ , $\boxed{\frac{3600}{7729}}$

• The numerator never changes and stay $3600$ but the denominator is trickier but we notice that:
• $\boxed{1897-1249=648=648\cdot 1}$
• $\boxed{3193-1897=1296=648\cdot 2}$
• $\boxed{5137-3193=1944=648\cdot 3}$
• $\boxed{7729-5137=2592=648\cdot 4}$
• We can conclude that : $\boxed{Rb_n=\frac{3600}{1249+648n\cdot \frac{n-1}{2}}=\frac{3600}{324n^2-324n+1249}}$
• Then $\boxed{Rb_{100}=\frac{3600}{3208849}}$ then $\boxed{e=3600}$ and $\boxed{d=3208849}$

• $\boxed{3208849-802489-900 \times 2026-9 \times 3600=550560}$

By Descartes’ Theorem , the radii $p_n$ of the purple circles are defined by $\frac{1}{p_n} = \frac{1}{16} + \frac{1}{p_{n-1}} - \frac{1}{25} + 2\sqrt{\frac{1}{16p_{n-1}} - \frac{1}{16 \cdot 25} - \frac{1}{25p_{n-1}}}$ , which with $p_0 = 9$ can inductively shown to be $p_n = \frac{3600}{81n^2+400}$ . Therefore, $p_{99} = \frac{3600}{81 \cdot 99^2+400} = \frac{3600}{794281}$ and $p_{100} = \frac{3600}{81 \cdot 100^2+400} = \frac{9}{2026}$ , so $a = 9$ and $b = 2026$ .

Also by Descartes’ Theorem, the radius of the $100^{\text{th}}$ green circle is $g_{100} = \cfrac{1}{\frac{1}{p_{99}} + \frac{1}{p_{100}} + \frac{1}{16} + 2\sqrt{\frac{1}{p_{99}p_{100}} + \frac{1}{16p_{99}} + \frac{1}{16p_{100}}}} = \frac{900}{802489}$ , so $c = 900$ and $d = 802489$ .

And the radius of the $100^{\text{th}}$ cyan circle is $c_{100} = \cfrac{1}{\frac{1}{p_{99}} + \frac{1}{p_{100}} - \frac{1}{25} + 2\sqrt{\frac{1}{p_{99}p_{100}} - \frac{1}{25p_{99}} - \frac{1}{25p_{100}}}} = \frac{3600}{3208849}$ , so $e = 3600$ and $f = 3208849$ .

Therefore, $f - d - c \cdot b - a \cdot e = 3208849 - 802489 - 900 \cdot 2026 - 9 \cdot 3600 = \boxed{550560}$ .