You are given 8 billiards balls, numbered 1 to 8, identical in shape and size. All but one weigh exactly the same. The odd ball is either heavier or lighter than the others. Using just three weighings on an ordinary two-pan balance, find the odd ball out as well as whether it is heavier or lighter than the others.
If you solve the previous puzzle, solve the same problem for 12 billiards balls (instead of 8) while keeping the maximum weighings to just three. Yes, there is a solution for this puzzle - if you give up, you can find the solution here.
For the first puzzle (with just 8 billiards balls), here is my solution (obfuscated using rot-13):
- Xrrc onyyf ahzorerq 1, 2 naq 3 ba bar cna naq 4, 5 naq 6 ba gur bgure.
- Vs gurl onynapr, gurl ner nyy abezny naq bar bs 7 naq 8 vf gur bqq onyy. Jrvtu 7 ntnvafg 1.
- Vs gurl onynapr, 8 vf gur bqq onyy naq lbh pna svaq vgf bqqvgl ol jrvtuvat vg ntnvafg 1.
- Vs gurl qba'g onynapr, 7 vf gur bqq onyy naq lbh jvyy vzzrqvngryl xabj vs vg vf urnivre be yvtugre guna gur bguref.
- Vs gur onyyf qba'g onynapr va fgrc #1, bar cna jvyy tb qbja. Jvgubhg ybff bs trarenyvgl, nffhzr guvf vf gur bar pbagnvavat 1, 2 naq 3. Rvgure bar bs 1, 2 naq 3 vf urnivre guna gur bguref be bar bs 4, 5 naq 6 vf yvtugre guna gur bguref.
- Xrrc 1 naq 5 ba bar cna naq 4 naq 2 ba gur bgure. Xrrc 3 naq 6 nfvqr.
- Vs gurl onynapr, bar bs 3 naq 6 vf gur bqq onyy. Sebz fgrc #5, jr pna pbapyhqr gung rvgure 3 vf urnivre guna gur bguref be 6 vf yvtugre guna gur bguref. Jrvtu 3 ntnvafg 1.
- Vs gurl onynapr, 6 vf gur bqq onyy naq vf yvtugre guna gur bguref.
- Vs gurl qba'g onynapr, 3 vf gur bqq onyy naq vf urnivre guna gur bguref.
- Vs gur onyyf qba'g onynapr va fgrc #6, rvgure gur cna jvgu 1 naq 5 jvyy tb qbja be gur cna jvgu 4 naq 2 jvyy tb qbja.
- Vs gur cna jvgu 1 naq 5 tbrf qbja, sebz fgrc #5 jr pna pbapyhqr guna rvgure 1 vf gur bqq bar bhg naq vf urnivre guna gur bguref be 4 vf gur bqq bar bhg naq vf yvtugre guna gur bguref.
- Jrvtu 1 ntnvafg 8. Vs gurl onynapr, 4 vf gur bqq bar bhg naq vf yvtugre guna gur bguref. Vs gurl qba'g, 1 vf gur bqq bar bhg naq vf urnivre guna gur bguref.
- Vs gur cna jvgu 4 naq 2 tbrf qbja va fgrc #10, sebz fgrc #5 jr pna pbapyhqr gung rvgure 2 vf gur bqq bar bhg naq vf urnivre guna gur bguref be 5 vf gur bqq bar bhg naq vf yvtugre guna gur bguref.
- Jrvtu 2 ntnvafg 8. Vs gurl onynapr, 5 vf gur bqq bar bhg naq vf yvtugre guna gur bguref. Vs gurl qba'g, 2 vf gur bqq bar bhg naq vf urnivre guna gur bguref.
- Va fgrc #5, vs 4, 5 naq 6 tb qbja, pneel bhg gur erznvavat fgrcf ol fjnccvat 1, 2 naq 3 jvgu 4, 5 naq 6 erfcrpgviryl.