Tactic trainer make a mistake!
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When you are doing problems in tactics trainer, there is a link in the upper right hand corner of the window that says "report problem". If you click this link, the staff will be alerted that there is a mistake in the problem. I recommend also putting your reason, including analysis, in the comments of the problem so that the staff can see it when they get your report.
The problem No 0023770 with an "official" sollution:
It's OK. However, look at a possition after the first move. It's an additionall solution: