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THE MERCHANT HAS 32 GOLD COINS. TO VERIFY THIS, DIVIDE THE 32 COINS INTO TWO UNEQUAL NUMBERS, SAY, 27 AND 5. THEN: 32 (27 - 5) = (272) - (52). 704 = 729 - 25 IF THE TWO UNEQUAL NUMBERS ARE 22 AND 10, THEN: 32 (22 - 10) = (222) - (102). 384 = 484 - 100 ALTERNATIVELY, LET'S SAY THAT THE 2 NUMBERS ARE X AND Y. WE CAN THEN COME UP WITH THE FOLLOWING EQUATION: 32(X - Y) = X2 - Y2 X2 - Y2 CAN BE EXPANDED TO: (X - Y)(X + Y) SO, NOW THE EQUATION WILL BECOME: 32 = X + Y (X + Y) WILL GIVE THE TOTAL NUMBER OF GOLD COINS FIRST HEAP: 22, SECOND HEAP: 14, THIRD HEAP: 12. TO SOLVE THIS PROBLEM, WE SHALL HAVE TO START FROM THE END. WE HAVE BEEN TOLD THAT AFTER ALL THE TRANSPOSITIONS, THE NUMBER OF MATCHES IN EACH HEAP IS THE SAME. LET US PROCEED FROM THIS FACT. SINCE THE TOTAL NUMBER OF MATCHES HAS NOT CHANGED IN THE PROCESS, AND THE TOTAL NUMBER BEING 48, IT FOLLOWS THAT THERE WERE 16 MATCHES IN EACH HEAP. AND SO, IN THE END WE HAVE: FIRST HEAP: 16, SECOND HEAP: 16, THIRD HEAP: 16 IMMEDIATELY BEFORE THIS WE HAVE ADDED TO THE FIRST HEAP AS MANY MATCHES AS THERE WERE IN IT, I.E. WE HAD DOUBLED THE NUMBER. SO, BEFORE THE FINAL TRANSPOSITION, THERE ARE ONLY 8 MATCHES IN THE FIRST HEAP. NOW, IN THE THIRD HEAP, FROM WHICH WE TOOK THESE 8 MATCHES, THERE WERE: 16 + 8 = 24 MATCHES. WE NOW HAVE THE NUMBERS AS FOLLOWS: FIRST HEAP: 8, SECOND HEAP: 16, THIRD HEAP: 24. WE KNOW THAT WE TOOK FROM THE SECOND HEAP AS MANY MATCHES AS THERE WERE IN THE THIRD HEAP, WHICH MEANS 24 WAS DOUBLE THE ORIGINAL NUMBER. FROM THIS WE KNOW HOW MANY MATCHES WE HAD IN EACH HEAP AFTER THE FIRST TRANSPOSITION: FIRST HEAP: 8, SECOND HEAP: 16 + 12 = 28, THIRD HEAP: 12. NOW WE CAN DRAW THE FINAL CONCLUSION THAT BEFORE THE FIRST TRANSPOSITION THE NUMBER OF MATCHES IN EACH HEAP WAS: FIRST HEAP: 22, SECOND HEAP: 14, THIRD HEAP: 12 WHEN THE FIRST SERVANT COMES IN, THE KING SHOULD WRITE DOWN HIS NUMBER. FOR EACH OTHER SERVANT THAT REPORTS IN, THE KING SHOULD ADD THAT SERVANT'S NUMBER TO THE CURRENT NUMBER WRITTEN ON THE PAPER, AND THEN WRITE THIS NEW NUMBER ON THE PAPER. LET X BE THE NUMBER OF THE MISSING SERVANT AND Y BE THE NUMBER THAT THE KING HAS WRITTEN. ONCE THE FINAL SERVANT HAS REPORTED IN, THE NUMBER ON THE PAPER SHOULD EQUAL: Y = (1 + 2 + 3 + ... + 99 + 100) - X (1 + 2 + 3 + ... + 99 + 100) = 5050, SO WE CAN REPHRASE THIS TO SAY THAT THE NUMBER ON THE PAPER SHOULD EQUAL: Y = 5050 - X SO TO FIGURE OUT THE MISSING SERVANT'S NUMBER, THE KING SIMPLY NEEDS TO SUBTRACT THE NUMBER WRITTEN ON HIS PAPER FROM 5050: 5050 - Y = X THE IDOLS FROM LEFT TO RIGHT ARE: GOD OF DIPLOMACY, GOD OF FALSEHOOD, GOD OF TRUTH. THE GOD OF TRUTH IS NOT SEATED ON THE LEFT BECAUSE HE ALWAYS SPEAKS THE TRUTH WHEREAS THE IDOL ON THE LEFT REPLIED THAT THE GOD OF TRUTH IS SEATED AT THE CENTRE. THE GOD OF TRUTH IS ALSO NOT SEATED IN THE CENTRE AS HE ALWAYS SPEAKS THE TRUTH BUT THE IDOL AT THE CENTRE REPLIED THAT THE GOD OF DIPLOMACY IS SEATED AT THE CENTRE. THEREFORE, THE GOD OF TRUTH IS SEATED ON THE RIGHT. AS GOD OF TRUTH IS SEATED ON THE RIGHT, AND HE ALWAYS SPEAKS THE TRUTH, THEN THE THE GOD OF FALSEHOOD IS SEATED AT THE CENTRE. THE GOD OF DIPLOMACY IS SEATED ON THE LEFT AND HE HAS LIED