Ponder This Challenge - August 2026 - The Wheel of Buttons

The Wheel of Buttons

This puzzle was suggested by Sanandan Swaminathan - thanks!

A game show is arranged as follows: The main attraction is a wheel that has $N$ buttons equidistant from one another, such that the wheel looks perfectly symmetric.

Each button has two possible states: "on" and "off". There's no indicator to the status of a button, and each press toggles between those two states ("on" -> "off" -> "on" and so forth). At the beginning of the game, the buttons have random states with the one condition that not all buttons are "on".

On each turn, the wheel spins randomly without the contestant looking; the buttons are marked with $1,2,\ldots$ beginning at the top-left button and going clockwise. The contestant picks a subset of the buttons and pushes them. After the contestant is done, two things can happen:

1. If all the buttons are "on", the contestant wins; this is made clear with balloons and confetti.
2. If not all the buttons are "on", a new round begins. Again, the wheel is spun without the contestant seeing it.

The contestant's goal is to find a strategy that guarantees winning in the least number of rounds, even if the game cheats and the spins of the wheel are not random.

The choices of the contestant can be described by a sequence of natural numbers. Rounds are separated by 0, and each round contains the numbers of the buttons pressed by the contestant.

For example, the following is an optimal solution for the case $N=2$:

If both buttons were "off" at the beginning, the game is won after the first round. Otherwise, after the second round both buttons have the same state, so after the third round, the contestant has certainly won the game.

Summing up the numbers in the solution, we arrive at the value $7$. If, in the middle round, we would have used 2 instead of 1, we'd reach the value $8$. We are interested in the minimum value that can be obtained from an optimal solution: We call this the "optimal solution sum" for $N$.

Your goal: Find the optimal solution sum for $N=8$.

A bonus "*" will be given for finding the optimal solution sum for $N=64$.

Solvers

• *Daniel Chong Jyh Tar (31/7/2026 4:54 PM IDT)
• *Jean-François Hermant (31/7/2026 5:28 PM IDT)
• Ashfaque Shaikh (31/7/2026 5:45 PM IDT)
• Alex Fleischer (31/7/2026 7:18 PM IDT)
• *Prashant Wankhede (31/7/2026 7:31 PM IDT)
• *Stéphane Higueret (31/7/2026 7:34 PM IDT)
• *Bertram Felgenhauer (31/7/2026 8:39 PM IDT)
• *Juergen Koehl (31/7/2026 9:51 PM IDT)
• *Paul Lupascu (31/7/2026 9:57 PM IDT)
• *King Pig (31/7/2026 10:38 PM IDT)
• *Dan Dima (1/8/2026 1:23 AM IDT)
• *Alper Halbutogullari (1/8/2026 2:16 AM IDT)
• *Jack Saleeby (1/8/2026 3:08 AM IDT)
• *Kang Jin Cho (1/8/2026 3:32 AM IDT)
• *Steffen Wolf (1/8/2026 4:48 PM IDT)
• *Rahid Zaman (1/8/2026 6:48 PM IDT)
• *Rethna Pulikkoonattu (2/8/2026 3:41 AM IDT)
• *Carter Tran (3/8/2026 9:49 AM IDT)
• Nadir S. (3/8/2026 11:01 AM IDT)
• *Lazar Ilic (3/8/2026 8:20 PM IDT)
• *Patricia Ji (4/8/2026 4:34 AM IDT)
• *Guangxi Liu (4/8/2026 5:10 AM IDT)
• *Jackson La Vallee (4/8/2026 6:29 AM IDT)
• *Tamir Ganor & Shouky Dan (4/8/2026 11:28 AM IDT)
• *Siamak Alimirzazadeh (4/8/2026 12:15 PM IDT)
• *Latchezar Christov (4/8/2026 2:10 PM IDT)
• *Franciraldo Cavalcante (4/8/2026 3:08 PM IDT)
• *Giorgos Kalogeropoulos (4/8/2026 5:46 PM IDT)
• *Abhishek Sekar (4/8/2026 6:01 PM IDT)
• *Vladimir Volevich (5/8/2026 3:39 PM IDT)
• George Jiri Spitalsky (5/8/2026 8:24 PM IDT)
• *Kevin Yu (5/8/2026 10:12 PM IDT)
• *Henk Waßmann (5/8/2026 11:09 PM IDT)
• Mark Buisseret (6/8/2026 8:40 AM IDT)
• Amos Guler (6/8/2026 7:30 PM IDT)
• *Evan Semet (6/8/2026 9:21 PM IDT)
• *Dominik Reichl (6/8/2026 10:51 PM IDT)
• Fakih Karademir (7/8/2026 4:00 PM IDT)
• *Sullivan Hart (7/8/2026 6:38 PM IDT)
• Baocheng Jiao (8/8/2026 1:10 AM IDT)
• *Aayaam Panigrahi (8/8/2026 9:12 AM IDT)
• Ahmet Yüksel (8/8/2026 12:51 PM IDT)
• Sakib (8/8/2026 1:33 PM IDT)
• *Harold Gutch (8/8/2026 2:27 PM IDT)
• Evert van Dijken (9/8/2026 11:05 AM IDT)
• Resnina Tathyana (9/8/2026 11:31 PM IDT)
• *Justin Mazenauer (10/8/2026 12:34 PM IDT)
• *John Tromp (10/8/2026 9:54 PM IDT)
• *Peter Moser (10/8/2026 10:39 PM IDT)
• *Paulo Sousa (11/8/2026 2:19 AM IDT)
• *Sven Schneider (11/8/2026 2:58 AM IDT)
• Daniel Bitin (12/8/2026 12:58 AM IDT)
• Shirish Chinchalkar (12/8/2026 2:59 AM IDT)
• *Lawrence Hon (12/8/2026 6:22 AM IDT)
• *Lucas Reymond (12/8/2026 1:22 PM IDT)
• *Tongyang Song (12/8/2026 4:23 PM IDT)
• Pataki Gida (13/8/2026 1:19 AM IDT)
• *Jason Shaw (14/8/2026 12:23 AM IDT)
• *Martin Thorne (14/8/2026 12:28 AM IDT)
• *Peter Ji (14/8/2026 5:39 AM IDT)
• Hakan Summakoğlu (15/8/2026 12:38 AM IDT)
• *Parth Rana (16/8/2026 7:47 AM IDT)
• *Nickita Khylkouski (16/8/2026 8:10 PM IDT)
• Karl D’Souza (16/8/2026 11:52 PM IDT)
• *Emek Can Doğru (17/8/2026 9:58 AM IDT)
• Dieter Beckerle (17/8/2026 10:45 AM IDT)