These are original reduced study positions, not saved games. Points are numbered from Light’s perspective, 24 down to 1. Unshown own checkers have already been borne off. In the bearing-off studies, Dark still has one checker outside Light’s home board. Before opening an answer, identify which rule restricts the tempting move.
In this guide
Exercise 1: the bar comes first
Light has one checker on the bar and one on point 6; Dark blocks point 24 with two checkers. The dice are 1 and 2. May Light begin 6–5, or must another move come first?
Position as text
Light 1 × 6; Dark 2 × 24. Light on bar: 1; each side has 13 off.
Show hint
The entry point determines which die can leave the bar.
Answer and explanation
First bar–23 using the 2. The 1 would enter on blocked point 24. No other own checker may move before re-entry. Afterwards the 1 can play 23–22 or 6–5, for example.
Exercise 2: only one die can be used
Light has one checker left on point 1; all others are off. The dice show 1 and 6. Which die must remove the final checker?
Show hint
Each die is individually legal, but both cannot be used in the turn.
Answer and explanation
The 6. At most one die can be used, and both are individually legal, so the higher takes priority. The oversized 6 is permitted because no own checker is farther away than point 1.
Exercise 3: oversized is not always allowed
Light has one checker on point 6 and one on point 3. With a single remaining die of 4, may the checker on point 3 be borne off?
Answer and explanation
No. While the checker on 6 is farther away, the 4 cannot bear off from 3 as an oversized die. The legal move is 6–2. If the only remaining checker were on 3, a 4 could bear it off.
Exercise 4: count a six
How many ordered results of two fair dice contain at least one six? List results with a six on the first die, then add only the missing results.
Answer and explanation
Eleven of 36. Six outcomes begin with 6; five more end with 6. Do not count 6/6 twice. Equivalently, subtract the 25 outcomes without a six from 36. This counts dice faces, not the legality of a particular hitting move.