Each of the six boxes
,
,
,
,
,
initially contains one coin. The following operations are allowed
Type 1) Choose a non-empty box
,
, remove one coin from
and add two coins to
;
Type 2) Choose a non-empty box
,
, remove one coin from
and swap the contents (maybe empty) of the boxes
and
.
Determine if there exists a finite sequence of operations of the allowed types, such that the five boxes
,
,
,
,
become empty, while box
contains exactly
coins.
Proposed by Hans Zantema, Netherlands
![B_1](/media/m/5/d/9/5d9518a7c0ead344571aac61b51bb25c.png)
![B_2](/media/m/1/8/1/181de00f42000a442a347ff370e521f1.png)
![B_3](/media/m/4/b/a/4ba38f08b828c32ed1e0c672a3bd9998.png)
![B_4](/media/m/1/2/a/12a398a25b8e58a722261566f8c92907.png)
![B_5](/media/m/e/7/4/e746548d8d04c2ba0d68a78491af7ea5.png)
![B_6](/media/m/8/9/d/89d7e2e6bfae2ff3bfab69706f73d15d.png)
Type 1) Choose a non-empty box
![B_j](/media/m/4/e/8/4e84bacb4fc45a510492f1137862fc72.png)
![1\leq j \leq 5](/media/m/6/b/1/6b1ccc46bfbf678cf34a5c5e2bd9dcab.png)
![B_j](/media/m/4/e/8/4e84bacb4fc45a510492f1137862fc72.png)
![B_{j+1}](/media/m/4/1/e/41ed885a95b6cc485d40eccf0bacc067.png)
Type 2) Choose a non-empty box
![B_k](/media/m/0/5/a/05a1ff80c1636fed2d57256950cac045.png)
![1\leq k \leq 4](/media/m/6/0/2/6023273b39452b0a4a9cef29b50400c0.png)
![B_k](/media/m/0/5/a/05a1ff80c1636fed2d57256950cac045.png)
![B_{k+1}](/media/m/8/5/e/85e19fb4708ea5a9631f1ae0c058a8b1.png)
![B_{k+2}](/media/m/e/c/d/ecd3876dea21d2d1816c3d6f97054bf7.png)
Determine if there exists a finite sequence of operations of the allowed types, such that the five boxes
![B_1](/media/m/5/d/9/5d9518a7c0ead344571aac61b51bb25c.png)
![B_2](/media/m/1/8/1/181de00f42000a442a347ff370e521f1.png)
![B_3](/media/m/4/b/a/4ba38f08b828c32ed1e0c672a3bd9998.png)
![B_4](/media/m/1/2/a/12a398a25b8e58a722261566f8c92907.png)
![B_5](/media/m/e/7/4/e746548d8d04c2ba0d68a78491af7ea5.png)
![B_6](/media/m/8/9/d/89d7e2e6bfae2ff3bfab69706f73d15d.png)
![2010^{2010^{2010}}](/media/m/3/7/a/37abbc4253cc537aa4bbb39e63abb884.png)
Proposed by Hans Zantema, Netherlands