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http://en.wikipedia.org/wiki/Otto_v...r.C3.A4sident_.28Prime_Minister.29_of_Prussia

The regent became King William I upon his brother's death in 1861. The new monarch was often in conflict with the increasingly liberal Prussian Diet. A crisis arose in 1862, when the Diet refused to authorise funding for a proposed re-organization of the army. The King's ministers could not convince legislators to pass the budget, and the King was unwilling to make concessions. Wilhelm threatened to abdicate (though his son was opposed to his abdication) and believed that Bismarck was the only politician capable of handling the crisis. However, Wilhelm was ambivalent about appointing a person who demanded unfettered control over foreign affairs. When, in September 1862, the Abgeordnetenhaus (House of Deputies) overwhelmingly rejected the proposed budget, Wilhelm was persuaded to recall Bismarck to Prussia on the advice of Roon. On 23 September 1862, Wilhelm appointed Bismarck Minister-President and Foreign Minister.

So WI the king had abdicated instead? Keep in mind that the crown prince didn't have a very good opinion of Bismarck and is unlikely to ever appoint him minister-president. We would have a non-Bismarckian and quite possibly liberal Prussia.
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